Is it a problem to have water around a tub drain flange?
However, after draining, there is a tiny bit of water around the flange, and I'm wondering if this is something I should be worried about. ...
  Will the standing water cause any problems.


It is not a problem if all other aspects have been implemented correctly.
Because: 

It's a "feature" of the geometry plus surface characteristics of the materials. The latter will change with time and may change the result (but it still doesn't matter regardless).

The "head" of water or pressure is so close to zero that the joint 'does not even know the water is there'.
In some circumstances you could get 'wicking' of fluid into a boundary between two materials by capilliary action - but a tub joint is designed to not be an example of this.

The water will evaporate in under a day in most atmospheric conditions.
Maybe not at about 0.5 degrees C and 99% humidity but in most others - and the materials used are designed to be permanently wet without other than cosmetic degradation.