Can I compare Shannon-Wiener diversity indices between plots of different sizes (area)?
Theoretically the index for two habitats of different areas are not directly comparable. Consider the species area relationship. You see more species when you look over a larger area. So you will potentially see a different index just from considering a larger area, and not necessarily because there is something different about the habitats. 

Borda-de-Agua et al tackle this head on. They use fractals to account for differences in area when comparing diversity indexes. It's complicated and I won't try to put the methods here because I don't understand them. Definietly worth looking at yourself though.

I would also consider using species abundance distribution (SAD) curves for comparison. They give information about diversity but don't sum it up in one nice number like indexes. I feel they are area invariant too. Since if you keep increasing the area of your sample, the total species will only increase from relatively few rare species, while abundances will increase mostly with the abundant species. This will not change the slope of the curve but just the intercept (My hypothesis, not sure if it's actually true). Checkout a good comparison of SAD's and diversity indexes in the intro to McGill et al. 



References

Borda‐de‐Água, L., Hubbell, S. P., &amp; McAllister, M. (2002). Species‐area curves, diversity indices, and species abundance distributions: a multifractal analysis. The American Naturalist, 159(2), 138-155. Link

McGill, B. J., Etienne, R. S., Gray, J. S., Alonso, D., Anderson, M. J., Benecha, H. K., Dornelas, M., Enquist, B. J., Green, J. L., He, F., Hurlbert, A. H., Magurran, A. E., Marquet, P. A., Maurer, B. A., Ostling, A., Soykan, C. U., Ugland, K. I. and White, E. P. (2007), Species abundance distributions: moving beyond single prediction theories to integration within an ecological framework. Ecology Letters, 10: 995–1015. Link