Don't know of any horizontal stablisers that can be trimmed to produce lift rather than down-force. That would imply the Centre of Gravity had moved behind the Centre of Lift, or at least very close to it, which might cause all kind of stability issues. Regardless, whether you're producing lift or downforce you're still generating induced drag. The trick is, obviously, to reduce downforce as much as possible, thereby reducing the induced drag from the stabiliser as much as possible, for the benefit of burning less motion lotion.
The real price to pay for a tail (or nose for that matter) heavy trim is a reduction in elevator trim available, and to a certain degree pitch authority.
As for tail-tipping, that's mainly a question of using the correct procedures during loading and un-loading. We'd regularly trim the MD-11F very close to the aft limit, target actually being right at the limit, and that aircraft is more prone to tail-tipping than anything else I've ever encountered (DOI around 96, it tips at index 100!). However, by following the established procedures it's possible to both load it for a very aft CoG and maintain ground stability.
The long of the short is that tail-heavy is beautiful. Doesn't matter if you're flying long, medium or short-haul; it all adds up and the potential cumulative savings are enormous. To give you an idea, the difference in fuelburn between an aircraft trimmed to forward most and aft most limits can be up to 5%.