Relation between order and stability in IIR filter
Stability depends on the pole locations, so one cannot generally say that increasing the filter order does or doesn't affect stability. It depends on how you increase the filter order. What are the additional coefficients of the denominator? (Because the denominator polynomial of the transfer function determines the pole locations).

Another question are the finite word length effects. If you quantize the filter coefficients, e.g., by using fixed point arithmetic (but of course also by using floating point), then higher filter orders tend to be more problematic than lower filter orders, because the chance that poles get very close to (or even outside) the unit circle through quantization becomes larger. But again, you cannot say anything in general unless you specify the filter's transfer function, the filter structure, and the type of quantization.