Excluding the 'dummy variable trap', are the problems from including too many interaction terms in a regression any different from the problems of including too many continuous or binary variables in a regression?
Furthermore, are the responses to this question conditional on interaction terms being continuous-with-continuous or binary-with-continuous?
In theory one column in the model matrix is much like another. However:
Interactions are in general less precisely estimated than main effects. Informally this is because the conditional means within each cell are based on fewer observations than for the main effects.
If they involve categorical main effects they can lead to unexpectedly high loss of degrees of freedom as people often fail to take into account that the product of the degrees of freedom for the main effects is involved so if you have an interaction between a categorical variable with 5 levels and one with 6 you lose $(5 - 1) \times (6 -1) = 20$ more df.
They may involve categorical main effects whose product implies the existence of empty cells with consequent difficulty in giving a simple scientific interpretation.
They are much harder to explain than main effects especially for three and four way interactions and for interactions involving two or more continuous variables.
