Do I need to replace my hot water tank's pressure release valve?
I would replace the P-T valve regardless, as it sounds questionable. They're about $10-15, plus usually a pipe directing it to the floor or even a nearby floor drain (which you can probably re-use). 

It's a safety device that prevents your tank from over-pressurizing (which in very extreme cases can actually cause an incredibly powerful explosion -- check youtube for some entertaining mythbusters clips). You also don't want it to trip accidentally, as it will pour a lot of very hot water out and will not stop until the P-T valve shuts itself off automatically (unlikely if it's broken), or someone shuts off an upstream valve. If your floor drain is clogged, or gets overwhelmed, you have a flooded basement/house.