We consider partially linear models of the form Y=XTβ + ν(Z) + ε when the response variable Y is sometimes missing with missingness probability π depending on (X, Z), and the covariate X is measured with error, where ν(z) is an unspecified smooth function. The missingness structure is therefore missing not at random, rather than the usual missing at random. We propose a class of semiparametric estimators for the parameter of interest β, as well as for the population mean E(Y). The resulting estimators are shown to be consistent and asymptotically normal under general assumptions. To construct a confidence region for β, we also propose an empirical-likelihood-based statistic, which is shown to have a chi-squared distribution asymptotically. The proposed methods are applied to an AIDS clinical trial dataset. A simulation study is also reported.