I have interest in my manuscript from a couple other presses, but my preferred publisher hasn't responded yet. If I let them know that they have some competition, would it get them to give me an answer sooner?
Well, depends. Answer these questions first:
1. Has your preferred publisher had the amount of time it requires (as posted in their submission guidelines) to evaluate your submission? If not, sorry. Trying to jump the queue is more likely to get your manuscript declined without a look than otherwise.
2. Not all publishers are in the same league. Would your preferred publisher call those other presses "competition"? If not, sorry again. Keep twiddling.
Ok. So let's assume that everybody has had enough time to review your manuscript, and all the publishers in question are (a) selling into the same markets and (b) of comparable reputation.
In this case and this case only, yes, you may inform your preferred publisher that press X and press Y are interested in the manuscript you submitted on date Z. Be sure to name names. If I get a letter informing me that unnamed publishers are interested in your manuscript, I will assume you're jerking me around and decline your manuscript without looking at it.