no you dont have to pay any gems at all....just clik on the gem itself in the slot and a window will pop up asking to accept the removal or decline...this is the way u can merge and embed larger gems.
When you try to remove a gem of higher than level 3, that very same window shows how much it will cost.
A level 4 gem (embed) requires 100 gems to remove
A level 5 gem (embed) requires 200 gems to remove
and so on