I run my hot water in the bathroom sink and the toilet next to it starts bubbling.
What could be causing this?
Bubbling of the toilet, along with gurgling sounds from drains, are usually signs of a blocked vent pipe. If your sink drains slowly at first, this is most likely the case. Unfortunately, clearing the vent often involves getting on the roof, removing anything visible on the vent pipe, and then clearing the vent with a garden hose or long snake.
It's possible that the sink is vented with an AAV, a special one way valve, that's installed as close to the top of the sink as possible, and if this fails, you'd have the same problem as a clogged vent. The solution in that case is to replace the AAV.
If there's a clog downstream of the toilet, as Netduke suggest, that would be evident if running the sink eventually backups up into the toilet or bathtub/shower. If that's the case, you can simply snake the line with a special snake designed for toilets or a long snake run from the sink.