Welcome, Aberlour!

I doubt the hot water heater would raise the temp enough inside that closet to affect your stored water. If your water heater is not vented properly, I suppose the exhaust gases could give the water a "taste" eventually, but if it's not vented properly, your emergency water is the least of your problems!

Actually, off topic here but I recently discovered that my water heater was not vented properly. I had a roofer do a bit of work on my garage roof and they must've knocked against the vent pipe that sticks up above the roof because the bottom end was no longer positioned above the exhaust port of the water heater anymore. And I only noticed it because my 16-year old water heater finally gave up the ghost. Who knows how long that condition could've gone on unnoticed if my water heater hadn't gone bad?

By the way, I'm not so sure it's necessary to add bleach when you fill up. I don't. I know it's a common thing to recommend, but seems redundant to me. I think it's more prudent to use bleach or other disinfection method at the time you use the stored water, just in case. It's highly unlikely that clean water, stored properly, will grow anything (unless you use untreated well water or something like that), but if there are any critters in there, you'll want to zap them at the point of use, not when you put it away.