Here is a bibliographic reference related to your question:
Wu, Lishan, "Characterization of protein dimerization equilibria by NMR diffusion measurements", Journal of the American Chemical Society 2009 Volume: 131 Issue: 34
I think it mostly depends on what type of molecule you are looking at. Can it participate in hydrogen bridging (e.g.alcohols, amides, sulfides), pi stacking (e.g. aromatics), or maybe even covalent bonding (e.g. disulfide formation). The possibilities are nearly endless.
Depending on the type of interaction, peaks may shift upfield, downfield, broaden, etc.
If the compound is an aromatic having pi-pi stacking interactions and also -OH group is present. What kind of shift is observed in the NMR spectrum for the aggregates?
If the OH is involved in hydrogen bonding, it is expected to shift downfield (higher ppm), at least if the NMR solvent is non-protic. Otherwise it will disappear. The OH signal may also become sharper. As for the effect of the pi stacking, I believe it should cause a shift to lower ppm values for the aromatic protons. Also, aggregation may cause a significant broadening of the signals.
In any case, unless dimer formation is very energetically favorable, most of your compound in solution should still be the monomer, with the dimer signals increasing in intensity with increasing concentration.
Along the aswer given by Lassaad I recommend to take a look at this publication Article Quantitative Interpretation of Diffusion-Ordered NMR Spectra...
The authors related diffsuionconstants of small molecules (not proteins) to their MW. Measurment of Diffusion in dependence on concentration might help to identify dimers. The group also has a Excel worksheet on their homepage which works nicely!
Please keep in mind that interpretation of diffusion for formation of dimers always requires a reference compound (e.g. solvent) for which diffusion is ONLY affected by the viscosity of the sample (which may also depend on the concentration of the solute) and not by "binding interaction" with other compounds!