Short answer is yes, according to my knowledge most linear regression models (even if used for classification) suffer from the "curse".
However, some regularised regression models are well suited for the case
features >> samples.
See Lasso (and its grouped and sparse variants) and elastic net. In general, what they tend to do is dropping many features (ie, many coefficients will be 0) so that, in practice, they do both feature selection and regression.
Have a look, for example, at scikit-learn guides and implementations to see what I mean.