Drawbacks of upsampling using polynomial interpolation
Linear and more generally  polynomials are pretty common methods for interpolation or extrapolation, easy to implement, and simple in a causal setting (prediction). Parabolic interpolation (in the Fourier domain)  or Savitzky-Golay filtering are practically useful examples, especially for peak-like signals.

Polynomials however can be  poor at modeling finite support data, and are subject to instability, oscillations or overshoot: they don't have finite bandwidth in general, so they don't cope with standard "sampled signals" assumptions.

Addition (triggered by Fat32's answer): when you fit a polynomial under a least-norm, potentially with a penalty term, the regression may get rid of noises or outliers, so there is indeed a filtering effect). I have experienced that, for piece-wise continuous signals, using several weighted polynomial interpolation in parallel, and choosing the best one locally can be pretty efficient, even-though it does not lead itself to easy theoretical derivations (CHOPtrey).