"I don't go to noisy places."
First of all, 行けません means "not able to go" whereas the original question uses 行きません, implying you choose not to go even if you could ;)
To answer your question, I don't think you can without changing the nuance of the sentence, but I'm not sure. I'm not a native Japanese speaker, but using 場所 somehow feels more specific and wouldn't seem as natural in this sentence. That is, うるさいところ means noisy places in general, but うるさい場所 means specific locations/venues that are noisy. (E.g. そんなにうるさい場所で仕事できますか。 / Can you work in somewhere as noisy as that?) Again, I could be completely wrong, but that's what I think the difference is.