Can I communicate without changing tones?
I've been also curious about this and I think this could easily be a research topic. Chinese people often happen to understand me even when my tones are flat, and I highlight that they understand me a lot less when my tones are wrong.
Of course this depends a lot on the context, and on the sentence. In a known context and with basic (high frequency) words tones are not that important (I would agree with @NS.X, 80/90%). The speech could sound weird to Chinese people but they can understand. Take also into account that they are used to a wide variety of accents from all China.
But, the grammar and the set phrase is better to be perfect.

If conditions are not so particular, tones are important as grammar and words, so recognition percentage could easily fall below 50% of the sentences.

I want to add that:


Sometimes I noticed for some words Chinese people have difficulty to remember which tone it is. For them is natural to put that tone on that sentence.
When they wisper, or better, when they speak crying (you can see sometimes in the movies) tones almost disappear.


About your study I can say tones are really important, you have to study them and try to learn, but I also say you that I memorized most of them just hearing and using them. You will end up repeating a sentence in your head and saying "Ok, wo is third tone".