I'm baking a genoise cake consisting of flour, eggs, sugar, and optionally some melted butter. I want to add a strong tea flavour to it.
I'm afraid adding tea directly will mess up the batter, so can I replace the sugar with a simple syrup made with tea? If yes, is there anything I should adjust - like removing the egg whites to reduce moisture?
No, you can't do that. It will totally change the whole cake layer and make it potentially inedible. Besides, the tea flavor won't be strong.
You cannot remove the moisture, because the genoise is pretty much moistureless anyway. The egg whites cannot be removed. Not only is their role the opposite - they make a drier cake, not a moist one - but also they have a huge structural role in creating the sponge. Without them, the result won't be anything even close to a genoise.
If you really insist on trying this, you will have to use a different recipe, something much simpler than a genoise that produces a heavier, denser cake. Look for something that uses the muffin method, these recipes are much less sensitive to changes. Also, the added taste will be stronger, because the cake will be both more substantial, and will contain more fat. Ideally, find a recipe that is already written for a liquid form of sugar, like corn syrup or honey, and use your tea syrup there.
You don't need to replace the sugar with syrup just to add tea flavour. Powdered tea leaves such as matcha powder can be used as a dry ingredient.
Matcha Cake is a kind of sponge cake, which sounds similar to your genoise cake in texture. The linked recipe uses matcha powder for the tea flavour.
