"I'm lovin' it"
Solution 1:
It sounds fine to me. (But I'm not actually a native speaker, and Indian English does have a reputation for using the progressive a lot.)
This is how I interpret "I'm loving it". (I've put back the 'g' because writing lovin' is too folksy for me.) I also assume "it" refers to something particular, like McDonald's or the food there. Also, I think it helps to consider the analogous sentence "I'm enjoying it".
English certainly uses stative: I love chocolate, or I enjoy classical music. This expresses your position on something. But suppose I'm at a play/concert/movie, and someone asks me what I feel, during the event. I may say "I'm enjoying it" or "I'm loving it (at this moment / so far)". This seems perfectly natural to me, though perhaps not to native speakers everywhere.
There's also a difference between "I love it" and "I'm loving it". If you love chocolate, it only says what you feel when the issue of chocolate comes up, but if you are loving McDonald's, it implies that you're there right now and/or are actively engaged in thinking about it and loving it. (Just as "I'm enjoying classical music [right now]" means more than "I enjoy classical music".) Presumably, "I'm loving it" is a subtle suggestion that you too, like the speaker of the slogan, ought to be at McDonald's right now and actually get on with the act of loving it, not merely think of it as a nice place to visit from time to time. :-)
Solution 2:
Since there have been no other similar answers so far: I am a native English speaker and it sounds perfectly natural to me. Cheesy, but natural in terms of grammar.
EDIT: @Martha expressed the same sentiment, but as a comment rather than an answer. If I had seen that, I would have left a comment as well. It amounts to more or less the same thing though.
Solution 3:
It does sound a bit odd, but it implies that you love some experience that continues over a passage of time, rather than an explicit thing. "I love the McDonald's cheeseburger" is something you can say while you are not presently eating one, and you are expressing love for an object. "I'm loving the McDonald's cheeseburger" implies that you are currently eating one and you are feeling this "love" throughout the experience.
It might be McDonald's way of marketing themselves as offering an experience in addition to selling food.