Why to remove skin and seeds of tomatoes when we puree them?
For whatever reason, the brand of tinned tomatoes I used to buy regularly had somewhat bitter-tasting seeds; the flavour was definitely present in pureed soups / sauces. 

I used to squeeze them all out by hand, but some still made it into my precious San Marzano tomato sauces. 

Then I found the perfect tool for skinning and seeding larger quantities of tomatoes: the food mill. 

For the next several years, I dutifully used my mill to remove seeds from my favorite brand of (already skinned) tinned tomatoes - boring, yes, but the bitterness was gone.

Sometime during this period, I switched to a different brand of tomatoes. Then I got bored of the food mill, and pureed them whole again... with no bitterness. 

The moral of the story? Eat some of the seeds! If they're bitter, get rid of them before you puree. If you don't really notice, then don't bother.

My food mill still makes great mashed potatoes. Or are they milled potatoes?