Word limit for paper after addressing reviewer's comments

I can now provide an answer to my own question (I hope this is alright): I recently submitted a revised article that was about 15% longer than the word limit to a small biomedical journal. The original version of the article kept to the word limit. In the rebuttal letter, I explained that the article had become longer due to the reviewers' question. This article was accepted without further comment from the editor.

So it seems that at least some editors do not see a problems with longer articles, as long as the original submission keeps to the limit. I suspect that the rules may vary greatly between disciplines and journals and that no journal will ever explicitly state "we do not care about length for revised submissions". My PI was very confident that the article would be accepted, so it may be advisable to ask an experienced colleague in the field for their opinion.

Note — my answer is from an author's point of view with experience publishing in IEEE Transactions-type journals. Your mileage may vary.

Are there any guidelines on how willing journals might be to accept papers above the word limit after revisions?

The guidelines for revisions are the same as those for the initial submission; that is, there are no separate guidelines for revisions.

Is word length still strictly enforced or is this mainly checked upon initial submission?

The guidelines I've encountered on manuscript length have dealt with page count specifically, not "word length." So, yes, the page count is strictly enforced on the initial submission and any subsequent revisions. In your case, there is probably a hard limit on word count for the journal you are submitting to regardless if it is a revision or not.

Would simply submitting and hoping for a quick response from the editor (in case this is unacceptable) be a good strategy? Or is this a bad idea since either it highly unlikely to be acceptable or has a risk of annoying the editor?

You can submit your over-length paper if it makes you feel better; however, the opportunity to annoy is always present when someone doesn't follow the rules. Having said that, the editors I've worked with in the past have been fairly quick to send back to me submissions that did not follow the guidelines.

For example, one journal I submitted to had just changed it's maximum page count while a paper of mine was undergoing the last stages of an internal review. I had not checked the page limit prior to submitting as I was already familiar with what the guidelines said regarding this matter (after all, I structured my paper to be compliant with the guidelines). Sure enough, the editor sent it back within a couple of hours.

In conclusion, there is a balancing act of sorts that goes in to revising papers such that they address referee comments sufficiently while maintaining compliance with the journal's guidelines on manuscript length, etc.