Tea towels can be ironed, they will keep taking on water as usual.
It is the teddy cloth towels that do not take up as much water when ironed.
My solution is quite simple, as I hate ironing:
Shake out the towel when it comes out of the washing machine, best if you can get it to make a whip sound.
Then pull the fabric straight, with your hands to start and then you drag your edge over the edge of a table or even the washing machine. You do that from one corner to the next, with the first two inches/five centimeters being stretched and flattened on that semi-sharp edge.
Hang to dry or put in the tumble dryer.
When dry, repeat the shake, the pull and the drag over the edge of the table.
This works best if you start with it when the towel is still new, as the creases will wear faster than the bits in between, making them more likely to stay.
If you do iron other things, you can use the iron on just the edge of your tea towels. You will not use that first little bit for drying much, so even if it makes it taking water less easy, it will only be a small bit that is affected.
And when you use that tea towel for drying dishes later, you can test whether it does change the amount of water it accepts.