Hi libertas,
If you are talking about a standard steel bedframe then the mattress would sink right through the gaps between the support rails to the floor and you would probably damage it and it would certainly void the warranty.
If you have a wooden bedframe then it would depend on the distance between the slats and on the amount of flex in the slats. A memory foam mattress is designed to be used on a solid non flexing support surface or foundation (not a box spring which flexes). I wouldn’t consider gaps that are more than about 5" wide and less would be much better to prevent the mattress from sagging into the gaps over time. Narrower gaps and stronger wood will also have less flex and provide a stronger and more evenly supportive support surface. If the gaps in a wooden bedframe are too wide then I would either add slats or I would use a thin bunkie board (see post #4 here) over the slats in the bedfame. If there is too much flex in the slats then you can make T-Slats (see here) to make them stiffer.
The support system under a mattress can certainly affect how it feels and performs and the durability of a mattress as well.
Phoenix