The duration of the effect stacks, meaning you can, for example, drink two Loot Drop Potions in a row to get 1 hour of increased drop rate, and both the Loot Drop and the Loot Tier effects can be active at the same time. However drinking multiple boosters will only increase the duration of the effect(s), it will not make the drop rates raise even higher.
The effect timer counts down whenever the player is active on the character that the Loot Booster was used on. Like XP Boosters, the timer pauses when the player exits the game or pauses (via /pause), and resumes when the player reenters the game or unpauses, and if the character dies any remaining time on the Booster is lost. With effect from Patch X.31.9.3 (Sep 2019) the boosters no longer deplete in safe zones (Nexus, Vault, Guild Hall, Bazaar, Daily Quest Room, Pet Yard).
If you have multiple character slots, the effect will only be active on whichever character used the Loot Booster.
Items affected by Loot Boosters drop in a Boosted Loot Bag, which replace all other bags. This bag will appear, meaning that the loot boost effect is still active, as long as the timer has not expired when an enemy dies.