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