The answer to
how much information are we losing by considering a temperature-independent forcefield in such systems?
is, to a first approximation, none. The reason is that the statement
transitions are strongly dependent on the temperature induced changes in the interaction of the system
is actually exactly backwards.
Temperature-induced phase transitions happen because a system's internal interaction energies do not change with temperature. Now, in the canonical ensemble, the probability difference between microstates is inversely exponential in the energy divided by the temperature. Thus, because interaction energies do not change with temperature, as temperature increases, the probability difference between lower- and higher-energy microstates decreases, so that less ordered phases become entropy-preferred relative to more ordered phases.
To make this concrete, consider lipid bilayer melting. A lipid bilayer is held in place because interactions between all of the lipids' tails, and interactions between the lipids' heads and water, stabilize the system, and the bilayer arrangement is an easy enough way to accommodate everyone's preferences.
But let's say we take a thermal sledgehammer to the system and heat it up to 1000 K. Well, then, every molecule's (on average) got the kinetic energy to bounce from its pot right to the roof and back. Who cares about alkane dispersion interactions and water dipoles and any of that jazz any more? Mind you, the interaction energies haven't changed -- but the thermal energies have rendered them irrelevant. You know as well as I do that we end up with a cloud of vapour, whether in real life or in a (proper) simulation.
Now, when you ask
Are there examples that provide instances of failure of these temperature-independent forcefields?
of course they fail. SPC/E water freezes at -58 °C and boils at 123 °C, which are clearly not the correct freezing point and boiling point. However, it does freeze, and it does boil, and nobody set out to say "hey let's make a model of water that freezes and boils" or inserted special switches at those temperatures to suddenly change the interactions involved.
Instead, the reason molecular dynamics models fail to predict the correct transition points of everyday substances is that the potentials used simply cannot be accurate enough. Molecular dynamics atoms are balls on sticks with point charges and two-term equations for forces. Real life atoms are infinitely more complicated. So the exact temperatures at which the interatomic forces lose to entropy will be "wrong" if the interatomic forces themselves are "wrong" in the first place. But when the interatomic forces have the correct qualitative features, then applying temperature correctly in a molecular dynamics simulation must yield qualitatively correct phase transitions. This whole matter of correct thermostatting is an entire field in itself, but it is almost never done by meddling with the force field parameters in an arbitrary way.