There are three options, the mRNA "carrier", such as the lipid particle, the mRNA itself, and of course the protein product.
The first part is hard to guess without knowing what's in those delivery methods.
mRNA, or RNA in general, can form secondary structures which in theory could affect the the cell machinery. Depending on the details this could produce side effects to the cells that take up the mRNA. This would probably be localized and a strong secondary structure may inhibit translation also, rendering the vaccine ineffective. This can be checked in silico and in vitro during design to some certainty.
The protein product I would guess is the most likely to cause problems. Essentially you can encode any protein in mRNA, so you could make the cells that take it in produce snake poison, for example, if you wanted to. Depending on how the virus protein is modified, and how it reacts with the cells/body when present apart from the rest of the virus it assembles with could bring all kinds of side effects. The severity probably depends on what exactly it interacts with, and if it's only local or it gets into the bloodstream etc.