dollar doesn't magically go in that exchange, it needs to go through a bank. and banks have blocked bitfinex accounts by the time this topic was created (I don't have any updates on how this has changed though). so no it is not possible.
some exchanges however have an option called "codes" like btc-e codes which you can buy off the market and peer to peer and use that to fund your account. you need to check bitfinex, if they have this option then you can use that. I have used this method a couple of times to deposit fiat on btc-e by buying btc-e USD code with my local currency.
also last time I checked they never had any problems with any of the cryptocurrencies. and you could deposit and withdraw all of them including bitcoin.
Pursuer,
Thank you so much for your help and detailed response.