No.
Consider three variables, $X$, $Y$ and $Z = X+Y$. Their covariance matrix, $M$, is not positive definite, since there's a vector $z$ ($= (1, 1, -1)'$) for which $z'Mz$ is not positive.
Population covariance matrices are positive semi-definite.
(See property 2 here.)
Sample covariance matrices - depending on how they deal with missing values in some variables - may or may not be positive semi-definite. If pairwise deletion is used, for example, then there's no guarantee of positive semi-definiteness.