Why do universities tolerate uploading papers on their websites?

Why the university officials do not take action over this copyright violation?

Because they can't. There is no way to tell whether a given electronic preprint violates the publisher's copyright-transfer agreement or not. Different publishers place different restrictions on authors' rights to redistribute their papers. Some allow posting pre-edited versions; some allow posting the official camera-ready version; some allow neither; some only require an exclusive publication license and leave copyright in the author's hands. These restrictions change over time, and may depend on whether the author paid an open access fee to the publisher. The only way to determine whether an electronic (p)reprint is posted illegally is to read the actual copyright-transfer/publication contract. But this contract is directly between the authors and the publishers; universities have no record of these agreements.

Because they don't have to. At least within the US, university web sites generally fall under the "safe harbor" provisions of the Digital Millennium Copyright Act, which requires copyright holder to request removal of the specific items to which they claim copyright.

Because the publishers don't care. Scary legal language notwithstanding, academic publishers in many fields have zero interest in pursuing legal action against individual researchers for posting copies of their own papers in violation of copyright transfer agreements. (I have heard this said specifically about ACM, IEEE, SIAM, and Springer, by people with connections inside each of those organizations.)

Because it's not in their best interest. Both universities and the public benefit materially from the public availability of research by their faculty, students, postdocs, and other researchers. Universities have no incentive—aside from a potential legal threat that they know will never materialize—to proactively censor that research. Many universities, and more recently many governments, have adopted open-access policies that either encourage or require their members to amend publication agreements and make their work publicly available.

Because researchers would revolt. Even if academic publishers started sending DMCA notices to universities, and even if universities required their members to take down copies of their papers, in violation of research community expectations, the people whose research is being censored would simply take their business elsewhere. Those publishers would receive fewer papers, and those universities would receive fewer graduate school and faculty applicants.

Are you sure those "lots of professors" you're talking about are infringing copyright holder's right? Because of the way you put it, your "question" sounded to me as if you were claiming that posting your own paper on your personal website was always copyright infringement no matter what, which is not true.

For example, here's an excerpt from Terms and conditions associated with the American Physical Society Transfer of Copyright Agreement:

The author(s)... ...shall have the following rights (the “Author Rights”):

...

*3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.

So, you can upload the final, published version of your paper published in, say, Physical Review Letters on your personal website. It's perfectly legal from any perspective.

Of course, not all journals have the exact same policy, so what kind of right you have as the author can vary greatly from publisher to publisher and maybe from journal to journal as well. For instance, unlike APS, IEEE allows authors to upload the final, published versions only if they chose the open access option by paying a fee. (Note that this doesn't mean IEEE forbids any online distribution of your results on your own. For example, you can still post your final accepted manuscript on your personal website without paying the fee. IEEE Transactions on Information Theory even encourages authors to upload your manuscript to arXiv when submitting to the journal.)

I'm not familiar with other publishers' policies, but as far as I know, many mathematics journals allow authors to upload final accepted manuscripts on their personal websites.

Could you substantiate your claim by providing links to the many examples you're sure are the kind of copyright infringement you're talking about? Maybe this is peculiar to my field, but I don't remember many instances of such copyright infringement and am having very hard time believing this is prevalent.

Also, assuming there are actually many such ilegal cases, what makes you think that universities ignore this problem for the reason that it looks less criminal? I don't see why this explanation is the most plausible.

It is legal for a copyright holder to do whatever the holder wants to do with the work. The issue at hand is just who owns the copyright. Copyright can be owned by multiple parties, and some of those parties can sign away all or part of their rights. Doing so does not limit the rights of the remaining parties. The authors of an article may sign away their rights, but they generally do not have the authority to sign away any copyright that belongs to their employer. In the US, and perhaps elsewhere, work you perform for your employer is also copyright to your employer. In fact, depending on the terms of your contract, you may have already completely transferred copyright of any work you perform as part of your employment (and academic research likely applies) to your employer. Thus, publishers may be asking you to sign away something that is not yours.

I am assuming that by top universities, you mean institutions like MIT. MIT encourages employees to fight total transfer of copyright by amending the copyright transfer agreement. The gist of the amendment is that MIT contends that the authors' copyright to the material is nonexclusive - it also belongs to MIT. Thus it cannot be completely signed away by the authors to publishing companies. MIT will extend its privileges back to the authors regardless of what the publisher intends. Thus:

The Author shall, without limitation, have the non-exclusive right to use, reproduce, distribute, create derivative works including update, perform, and display publicly, the Article in electronic, digital or print form in connection with the Author’s teaching, conference presentations, lectures, other scholarly works, and for all of Author’s academic and professional activities.

and

Once the Article has been published by Publisher, the Author shall also have all the non- exclusive rights necessary to make, or to authorize others to make, the final published version of the Article available in digital form over the Internet, including but not limited to a website under the control of the Author or the Author’s employer or through any digital repository, such as MIT’s DSpace.

and

The Author further retains all non-exclusive rights necessary to grant to the Author’s employing institution the non-exclusive right to use, reproduce, distribute, display, publicly perform, and make copies of the work in electronic, digital or in print form in connection with teaching, digital repositories, conference presentations, lectures, other scholarly works, and all academic and professional activities conducted at the Author’s employing institution.

Since these rights are non-exclusive, they don't prevent the publisher from providing high quality print and online versions of the article to their subscribers. They also do not prevent the authors and MIT from making the article publicly available in a noncommercial way.

While MIT has taken steps to explicitly assert its rights to do so, many institutions likely have such rights implicitly (for example, they may be stated in employee contracts) and see no need to limit or infringe upon their own rights.