While people will often be dismissive and give answers like "of course not, you idiot", there's more to it than at first sight.
Before an answer can be given for this question, you need to ask yourself who is providing the WiFi. There are three possibilities here:
This is probably the only case that most people think of when they're giving a dismissive answer. In this setup, you're using a wireless router or a wireless access point to distribute your home broadband internet among several devices such as your desktop computer, laptop, possibly a wireless VoIP phone, and your WiFi-enabled mobile phone.
In this instance, your mobile phone is just another device connected to your wireless home network. Any data usage is already paid for in your broadband internet bill. While we all know that mobile network operators would love to bill you even just for thinking about using the phone, they have no way of knowing that you're using your own wireless network at home. There's no way they can bill you for usage they don't even know about.
So, in this instance, the answer is an emphatic "no".
2) Your mobile operator
Some mobile network operators (T-Mobile UK for one) have a network of publicly visible WiFi hotspots in places like pubs, fast food outlets, railway stations, airports etc. While they may be visible to all and sundry, not anyone can actually use them. They're protected by a login and password that you're not given until you pay for them.
Access can be flat-rate or PAYG. In the former case you pay the same amount whether you use the service or not, regardless of how much you use it (within the limits of a fair use policy established by the operator). In the latter case you pay according to how much data you use, either as an addition to your mobile bill or in the form of pre-paid credit.
So, in this case, the answer is an emphatic "yes". Your operator will bill you for using WiFi.
3) Third-party providers
Go into any "Varsity" pub in the UK and you'll be in range of a WiFi hotspot. Best of all, it's absolutely free. You can use their wireless network for free while downing a pint. Great for photographing or filming your mates doing silly things with bottles and sticking it up on flickr, Facebook, YouTube or whatever before they have time to say "whoops!"...
Another place that offers free WiFi if you stay with them is "Premier Inn" hotels. Once you've checked in you are given the network's ESSID and preshared key enabling you to connect to the network.
Not all such third-party providers allow you to use their wireless network for free, though. Some will charge. If you do have to pay to use the WiFi network, you will be billed directly by the owner or, in some cases where the owner and your operator have billing agreements, by your mobile operator.
In this case, the answer is a rather evasive "maybe, it depends on the owner of the WLAN".
Don't be dismissive. The answer is not always "no". It can be "yes" in certain circumstances!