  This article is concerned with a semilinear time-fractional diffusion
equation with a superlinear convex semilinear term in a bounded domain $\Omega$
with the homogeneous Dirichlet, Neumann, Robin boundary conditions and
non-negative and not identically vanishing initial value. The order of the
fractional derivative in time is between $1$ and $0$, and the elliptic part is
with time-independent coefficients. We prove
  (i) The solution with any initial value blow-up if the eigenvalue $\lambda_1$
of the elliptic operator with the minimum real part is non-positive.
  (ii) Otherwise, the solution blows up if a weighted $L^1$-norm of initial
value is greater than some critical value give by $\lambda_1$.
  We provide upper estimates of the blow-up times. The key is a comparison
principle for time-fractional ordinary differential equations.
