Is there any disadvantage to having open-access publications?

No. The important point is whether the journals are good (= publish good papers) rather whether the access is open or restricted. This being said, in many fields AFAIK the better journals are access-restricted.

I see this question is old and has been answered, but I would like to add that the accepted answer by Alexander is really just an opinion, and in my opinion, there can be disadvantages for publishing in open access journals. I'm not arguing against open access, but the disadvantages should be mentioned and taken into consideration.

The biggest disadvantage, in my opinion, is that some people consider open access journals to be a dumping ground for mediocre or crap science. These may or may not be the same people who are reviewing your CV or tenure. This may or may not be more true at older, more prestigious institutions, and/or older, stuck-in-their-ways scientists/academics. Some people even consider publishing in open access as career suicide.

With all that being said, I am a fan of open access, and think science should be accessible to everyone. "What if one has majority of his publications on open access journals?" I plan to publish my next manuscript in open access, but I will limit it to just one for awhile. Having the majority of publications in open access may throw up a red flag to some people. Of course, others may appreciate more open access publications. I am PhD student, and need more 1st author publications, but I don't want too many open access journals on my CV (for now).

Other things you might want to consider when choosing a journal:

• Cost is a major concern, and is one of the biggest reasons why I will go open access with the next paper.
• Impact factor is another reason. Many open access journals have a large readership and high impact factor, which is good. However, impact factor should not be your main goal.
• Citations should be your biggest concern. It's not enough for people to read your papers, you need them to cite you. And for that to happen, you need to have quality science and writing. Although a high impact factor may help with that.

Some funding agencies require to publish in open-access journals, or at least strongly encourage it. For example, I am funded by a funding agency that strongly encourages to do so. Unfortunately, in many fields, there simply aren't any high-quality open-access journals around.