What is the consequence of using an ordinary linear regression model while a linear mixed effect model is appropriate? Will the estimates and inferences be misleading?
We often use mixed models to account for non-independence of observations, which is one of the usual assumptions of linear regression. This often occurs due to clustering of observations, such as repeated measures within subjects. When observations within a subject are more similar to each other than to observations in other subjects, we will have non-independence so we fit random intercepts for subjects to acount for this. If we do not fit random intercepts, or account for this non-independence in other ways (eg with fixed effects for subjects, or with generalized estimating equations) then point estimates will still be unbiased and consistent, however standard errors will be wrong.