Here is the full list of all 58 Day of the Tentacle Remastered achievements. It takes around 5-6 hours to unlock all of the achievements on Windows.
Ask the IRS agents if they're related
Power Hoagie's Chron-o-John
Get a job
Make a skunk
Win the human contest
Disqualify Harold
Pick up 45 things that don't belong to you
Try to wash the dirty car with the brush
Red makes a super battery with only a little help from Hoagie
Get all six novelty items
Make vinegar
Deflate Oozo
Try to mix the red paint and the Booboo-B-Gone
Make it rain
Hoagie eats a Hoagie
Laverne tries the chattering teeth on her human during the human show
Enter the human contest
Pick up 30 things that don't belong to you
Microwave the frozen hamster
Try to flush the hamster
Talk to Dead Cousin Ted with all three kids
Clothe Laverne in the costume
Change the flag
Climb through the chimney with all three kids
Destroy a cherry tree
Create a cherry tree
Close a door
Try to alter the contract with Booboo-B-Gone
Open the safe
Try to mix the decaf and caffeinated coffees
Pick up 15 things that don't belong to you
Play Maniac Mansion
Try to mix the disappearing ink with Booboo-B-Gone
Find Fred's basement workshop
Get all other achievements in the game
Power Laverne's Chron-o-John
Acquire a diamond
Defeat Purple Tentacle yesterday
Play the whole game without skipping a cutscene
Sit through the end credits
Try to sign the contract yourself with the quill pen
Flush 50 objects through time
Talk to Washington, Jefferson, or Hancock while they're outside
Hoagie uses battery with Chron-o-John instead of plug
Wake up Dr. Fred
Get a tiny sweater
Try to microwave the hamster again after he's thawed out
Vacuum up a hamster
Get Fred to sign the contract
Reach yesterday without turning the stereo off
Return the stamp album to Weird Ed
Turn off the television
Squirt five different people with disappearing ink
Try to swap the textbook with the stamp album
Laverne thinks about eating the cat
Try to put something metal in the microwave
Ring the front desk bell 100 times
Cook the hamster in Maniac Mansion (Don't try this at home, kids!)