Can I accelerate caulk drying by using a fan or heater on it?
The tube of caulk should include drying and curing times.  It will vary based on the product so you definitely want to go by the manufacturer's recommendation and not some general guideline on the web somewhere.

You are not the only person who does not want to have to wait a day or two before using their sink and bathtub and the caulk manufacturers know this. It should be easy to find caulks that dry much faster than this.  Bathroom caulks are frequently advertising their "shower ready" time which is how long before it can get wet.  It may take a day or two to fully cure but many caulks will be able to get wet after a few hours.

Like many "fast drying" products the downside is that you have less time to work with the product before it starts setting up.  The last time I caulked my bathtub I used a very fast drying product that claimed it was "shower ready in 1 hour".  I am not a perfect caulker so I kept adding more and smoothing it out; unfortunately it started "clumping up" after maybe 20 minutes so I ended up with lumps.  If I had used a product that took longer to dry I would have had more time to smooth it without it starting to dry and get a skin over the top.