However, even if you were to use your current passport with ILR to travel in and out of the UK, you would not be able to renew your ILR sticker when it came time to renew your passport.
You could, completely legitimately, elect to have a Certificate of Entitlement to the Right of Abode placed in your current passport: British citizens who are also citizens of another country can have a certificate of entitlement in this way, provided that they don't have a British passport. This certificate confirms that you have the right of abode in the UK, and you would be able to use the UK/EU/EEA/Swiss entry points at airports, since you would not be subject to immigration control.
But the passport needs to be a valid one – as Jambo says, if you are looking not to get a British passport because your country of current citizenship won't allow you to keep it if you naturalise as British, it won't help you very much, since you'd be trying to travel using a passport that is no longer valid.
Beyond that, it is hard to see why you would want to do this. What is your country of current citizenship?
Of course, if you're not intending to travel at all, it's not necessary to have any passport!