Disadvantages

Linear regression models can only represent linear relationships, i.e. a weighted sum of the input features. Each nonlinearity or interaction has to be hand-crafted and explicitly given to the model as an input feature.

Linear models are also often not that good regarding predictive performance, because the relationships that can be learned are so restricted and usually oversimplify how complex reality is.

The interpretation of a weight can be unintuitive because it depends on all other features. A feature with high positive correlation with the outcome y and another feature might get a negative weight in the linear model, because, given the other correlated feature, it is negatively correlated with y in the high-dimensional space. Completely correlated features make it even impossible to find a unique solution for the linear equation. An example: You have a model to predict the value of a house and have features like number of rooms and size of the house. House size and number of rooms are highly correlated: the bigger a house is, the more rooms it has. If you take both features into a linear model, it might happen, that the size of the house is the better predictor and gets a large positive weight. The number of rooms might end up getting a negative weight, because, given that a house has the same size, increasing the number of rooms could make it less valuable or the linear equation becomes less stable, when the correlation is too strong.