What are some of the serious consequences that someone can suffer if they omit part of their academic record on their application for admission?

In addition to being denied admission or dismissed from the university, as noted in Bob Brown's answer, lying on an application can be grounds for revoking a degree after it was awarded. You may also be required to pay back any scholarships that were awarded to you under false pretenses.

For example, here's the policy of University of Arkansas at Little Rock:

If the false, misrepresented, or misleading information on the application portfolio is discovered:

• Before the application process is completed, the application will not be processed and no admission offer will be forthcoming.
• After admission and prior to enrollment, the admission offer will be rescinded.
• After admission and enrollment, the student will be administratively withdrawn from all classes and dismissed from the Graduate School; institutional financial assistance will be terminated retroactively and all tuition and other awards made to the student must be repaid.
• After a degree or certificate has been earned, the degree or certificate will be revoked.

Under no circumstances will the applicant receive a refund of his/her application fee or tuition and fees.

Finally, from the rest of the Internet, some anecdotes (I cannot vouch for their accuracy).

Via Powers v. St. John's Univ. Sch. of Law:

After David Powers had completed three semesters as a part-time law school student at St. John’s University School of Law, the law school discovered that Powers had made material misrepresentations and omissions in his law school application regarding his criminal history. St. John’s subsequently rescinded Powers’s admission based on the application’s material omissions and misrepresentations.

Via Poets and Quants: "Stanford Takes Away An MBA Degree":

For the first time ever, Stanford University’s Graduate School of Business today (March 5) confirmed that a 2003 graduate of the most selective MBA program in the U.S. no longer has a degree because he was admitted under “false pretenses.”

Via Yale Daily News:

He is Akash Maharaj, once a junior in Morse College, now a convicted felon. Kicked out of Yale College in the summer of 2007 for forging his application, he now faces five years of probation during which he must pay Yale over $31,000, for scholarships he stole — or else serve three years in jail.

Via The Right of Educational Institutions to Withhold or Revoke Academic Degrees:

The Supreme Court of New Jersey revoked the license of John Benstock to practice law in the State of New Jersey after New York Law School revoked his Juris Doctor degree for failing to reveal material information on his application to law school and admission to the bar.

Via hermitosis on MetaFilter:

I tried this once. I got accepted to the university and had forgotten to include a transcript from a summer community college course that I'd taken as a freshman in high school. I genuinely forgot to include it.

Anyhow I got a call saying that my acceptance was revoked, and when I visited the administrative office to try and explain, they accused me outright of deliberately covering it up in order to improve my chances of getting into their university (the grade had been a C, which lowered my overall GPA by just a smidge).

I was allowed to appeal their decision, but my appeal was rejected. And that's the story of how I was kicked out of university before I ever got to attend a single class (but not, I'm happy to say, before I'd already gotten my school ID, so I did wind up being able to draw upon some benefits).

Via happymomof1 on College Confidential:

I was almost thrown out of a graduate program that I was halfway through when the university realized that one (yes 1) transcript was missing from my file. In fact, even after it was delivered to them, and recorded as arriving, they managed to lose it again and I had to have it sent again. We are talking about a 30 year old transcript for 8 undergraduate credits taken in summer school! When I asked why they cared, the registrar (not a filing clerk) himself told me that missing transcripts can mean lost accreditation. No college or university wants to lose accreditation, they'd rather lose a student (or potential student).

Via airborne911 on College Confidential:

I attended two schools in spring 2006, but I forgot all about the second school (I only took one class there, and the rest of my classes were taken at the first school). I did not list the school on my application. USC sent me a request for transcripts from the second school in April. I called the admissions office to inform them that I simply forgot about the second school I attended. The lady on the other end of the phone said, "That's okay. That's what the National Student Clearinghouse is for." She then told me to send a detailed letter explaining why I didn't list the second school on my application, because omitting information is grounds for automatic rejection.

In the United States, and probably elsewhere, a student who lies on an application for admission can be denied admission or dismissed from the university, even after acceptance and starting classes or other work.