The engine manufacturer's minimum warm up time and the minimum oil temperature limit together are intended to ensure several things, as tdracer has discussed. My understanding is that the main intent is:
1) Minimum oil temperature for the lubrication to be adequate for the bearings to roll properly and withstand the loads of high power operation
2) Minimum oil temperature to ensure that, when fuel flow goes way up at power set, the temperature of the fuel downstream of the fuel oil heat exchanger does not dip below the freezing temperature of water for longer than the servo circuits of the engine control system hydromechanical unit have been demonstrated to withstand, and
3) The engine case has warmed up enough so that, when the centrifugal forces of high rotational speed expand the rotor at power set, the engine does not experience rub at the blade tip seals or stator and nozzle seals because the case hasn't yet expanded enough.
4) The rotor has warmed up and expanded enough so that, when the takeoff roll progresses and the case heats up and expands quickly, the clearances don't get too large a minute or two later before the rotor thermally starts catching up to the case, which can cause a stall around rotation or initial climb.
There is a combination of thermal effects, pressure effects, and centrifugal effects that determine the radial clearances between the rotor and the static parts as changes in cycle temperature occur. The rotor drum and disks have more thermal mass than the engine case, and they respond to changes in cycle temperature more slowly. On top of the effect of thermal expansion and contraction of the rotor (which lags behind cycle temperature changes), the centrifugal force on the rotor causes it to immediately grow and shrink with changes in rotational speed. The case grows and shrinks with changes in cycle temperature more quickly than the rotor (and turbine case cooling systems obviously also affect the turbine case). There is also a pressure driven expansion of the case, but I have understood this to be a relatively small effect compared to the thermal effects (maybe tdracer can confirm this). If you set takeoff power before the case has had a chance to warm up and expand some, you may get rubbing of the blade tips seals and stator and nozzle inner seals right at initial power set when the rotor expands immediately from centrifugal force. Once takeoff power is set, the case expands from the temperature increase more quickly than the rotor, and the maximum clearances in the compressor can be right about at rotation or initial climb. As tdracer said, if the clearances get too large or are not uniform, a compressor stall can occur.