I've multiple wallets and I was wondering if I keep transferring bitcoins among these wallets (my own wallets ) will that eventually consume all my coins (keep paying fees )?
Obviously, yes, it will, due to every transaction need a transaction fee. Not unless if you use a wallet such coinbase and send to another coinbase wallet which are hosted web wallets that never as ask any fees in transferring funds.
But for what reason you would do that?
Technically, I am not sure the user will run out of coins completely. I depends on the fees and the initial amount, but if the user is left with an amount that is not even enough to cover transaction fees, this person is kind of stuck with it. Well, of course, one can try to send it nevertheless and then the transaction might be successful once the fees get smaller, but then this process of getting rid of the last tiny bits of coins can take a really long time. Unless, of course, the person customizes the fee and sets it to the exact amount of what's left in the wallet.