When you pick up your cup, do it by the very bottom, and then use your pinkie to push whatever the cup is sitting on away from the cup.
You'll leave the napkin/coaster/placement on the table, and your cup will be in your hand.
Note that this works both right- and left-handed (try it; it's hard to pick up a cup with your pinkie on top).
lifehacks.stackexchange.com/users/7247/Daniel Griscom
Get into the habit of picking it up by tilting it slightly first.
The seal then breaks at the side that lifts first, and the mat doesn't stick.
lifehacks.stackexchange.com/users/19064/ChrisA
Sprinkle a little salt or other small solid particles on the mat before using it.
This can prevent the seal from forming between the condensation/residual moisture and allows you to freely use the cup until it's all dissolved.
lifehacks.stackexchange.com/users/8157/Flexo
Don't let yourself get in the situation from the start.
I'm sure you must have a something... a tea-towel, handkerchief, tissue, kitchen roll, trouser-leg, etc.
near to hand.
Dab the bottom of the cup before you leave the kitchen.
Your mat will look nicer for longer too.
there's nothing worse than a sticky old place-mat, that the owner doesn't notice...
yet everyone else does.
lifehacks.stackexchange.com/users/13777/Tetsujin
Add weights to the placemat, or use a heavier placemat.
For example, you could glue a small tile to the bottom of a rectangular placemat to weigh it down.
lifehacks.stackexchange.com/users/17627/Batman
I always make a point of placing the cup so that one very small part of the rim is OFF the placemat, then there's an air gap and no vacuum by which to lift the placemat with the cup.
lifehacks.stackexchange.com/users/1965/Lefty
If I ever have a drink and there is a bit of moisture on the bottom of the cup, then it sticks to the placemat, often without me knowing.  Then when I lift the cup up to take a drink, the placemat lifts up as well. Sometimes I notice, and it's just annoying, other times it will fall on the floor or in my food.  How can I prevent this?
How to stop cups sticking to placemats
lifehacks.stackexchange.com
<food>
lifehacks.stackexchange.com/questions/14800
lifehacks.stackexchange.com/users/18753/Beastly Gerbil
There are number of ways to avoid this problem, one of which is to pick up the cup in such a way that you can use your pinkie to push the placemat away from the cup. You could also pick up the cup by tilting it slightly first or use a heavier placement. To avoid the situation developing in the first place, you could sprinkle a little salt on the placemat first, place the cup so that a very small part of the rim is off the placement, or dab the bottom of your cup before you leave the kitchen. The options are to pick up the cup from the bottom and push the mat away with your little finger, tilt the cup slightly as you pick it up, sprinkle some salt on the mat before using it, Wipe the bottom of the cup before leaving the kitchen, use a heavier mat or leave a bit of the rim of the cup off the mat.
Pick the cup up from the bottom and push the coaster away with your little finger. Tilt the cup slightly as you pick it up. Sprinkle a little salt on the coaster before use. Wipe the bottom of the cup before leaving the kitchen so that the stickiness never forms. Use a heavier coaster. Place the cup so that a small part of the rim is off the mat.