I've come across an upsampler that uses polynomial interpolation and no filter. What are the drawbacks to this? I looks more efficient than filtering.
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).
If you are upsampling a class of bandlimited signals, such as specch, using polynomial interpolators, then you are introducing erros into your results. This is because polynomial interpolators will not have the necessary conditions for being an image free upsampler. The remaining spectral images are observed as errors in the interpoated signal.
However, if you are upsampling a class of non-bandlimited signals, such as computer generated graphics, then a polynomial interpolator may even have an advantage of preserving pixel-sharp edges which by definition have an infinite bandwidth and cannot be represented by a bandlimited signal model.
And furthermore, polynomial interpolators may also have an additional advantage of noise reduction if the signal of interest fits into a polynomial model while the added noise does not.