You can clearly see that currently Google is still using the real PageRank for the ranking.
I'm sure that they have alternative ranking algorithms without using PageRank data. But the answer to the original question is yes, removing PageRank completely will have effects. (Even if the impact decreased the last years.) And the effect for ranking will be more important than that for removing data from the toolbar.
Anyhow, I believe that Google and the Stanfort University will find an agreement. As long as there isn't any agreement Google won't spend time in PageRank, i.e. update the toolbar data.