The only situation I can see where you'd be paying BOTH a mortgage (of any type) AND rent on a property was if you'd bought it under a shared ownership scheme. Assuming that's the case, some figures might things clearer:
Let's say that you bought a house worth £200k under a 50/50 shared ownership scheme, using a £10k deposit. You'd own half the house (valued at £100k) and need to take out a mortgage for £90k in order to pay for the rest of that half. (That mortgage could either be a conventional one or an 'interest only one'). The other half (£100k) would be owned by the property developer (or whoever they sold it to), for which they'd charge you rent.
If (using my sample figures), you've simply converted an interest-only mortgage on £90k into a standard mortgage against the same sum, the property developer (or their successor) will still own the other half of the property, meaning that you'll still need to pay rent on it. [I'm guessing that's what's happened in your situation]. In order to eventually own the whole of the property, and not to pay rent, you'd need to convert your interest-only mortgage on £90k into a standard mortgage on £190k, allowing you to buy the developer's share of the house.
If you've already bought the developer's share of the property (and have got a mortgage to cover that purchase, as well as your original one) then, clearly, you shouldn't be paying rent.