Watching programmes on iPlayer requires a TV licence. That's the only payment you need make to the BBC.
Your BT phone bill, per se, is unaffected but you might get charged for the data you've used by your ISP. (That could be BT but it could equally likely be Plusnet, TalkTalk, Post Office, Sky, Claranet or any one of many other companies). That's unlikely these days, as most broadband packages offer unlimited data but people still using 'capped' services might get charged if they use more data in a month than their contract provides for.