Covariate-adaptive randomization is popular in clinical trials with sequentially arrived patients for balancing treatment assignments across prognostic factors which may have influence on the response. However, existing theory on tests for treatment effect under covariate-adaptive randomization is limited to tests under linear or generalized linear models, although covariate-adaptive randomization has been used in survival analysis for a long time and its main application is in survival analysis. Often times, practitioners would simply adopt a conventional test such as the log-rank test or score test to compare two treatments, which is controversial since tests derived under simple randomization may not be valid under other randomization schemes. In this article, we prove that the log-rank test valid under simple randomization is conservative in terms of type I error under covariate-adaptive randomization, and the robust score test developed under simple randomization is no longer robust under covariate-adaptive randomization. We then propose a calibration type log-rank or score test that is valid and robust under both simple randomization and a large family of covariate-adaptive randomization schemes. Furthermore, we obtain Pitman's efficacy of log-rank and score tests to compare their asymptotic relative efficiency. Simulation studies about the type I error and power of various tests are presented under several popular randomization schemes.