Will a professor get annoyed with frequent requests for recommendation letters? Just last month I asked three professors to write me recommendation letters for my graduate school application. They all agreed. Then today I found an internship that also requires recommendation letters.

I am interested in applying to the internship, but I worry whether I would be asking for letters from my professors too frequently, especially since they might need to rewrite the letters to accommodate the nature of an internship.

A related post here, although in that case the two requests are one year apart and the letters are for the same purpose.
It is fine; don't worry about it.

This is a very common situation: many graduates will apply to more than one program that needs recommendation letters.  They almost certainly are going to be asking the same few people for multiple letters, because they are unlikely to have a large number of contacts to draw on to ask for recommendations.

I think the question you have linked very much applies to this case.  Particularly the top answer, which points out that this is part of professors' jobs, not some kind of exceptional favor you are asking for.