Does multiplying the likelihood by a constant change the Bayesian inference using MCMC?

For numerical Bayesian inference we have Posterior~Prior*Likelihood. In MCMC we do not need to calculate the denominator in Bayes rule. My question is that can I multiply the Likelihood by a large constant while making sure that the posterior is still the same?

When you multiply the likelihood by the prior, the resulting function may no longer integrate to $1$, hence why you need to know the normalising constant to analytically solve the posterior.
Not only is that allowed, but the multiple you get is still a valid likelihood function. A valid likelihood function is defined by the requirement that $L_\mathbf{x}(\theta) \propto p(\mathbf{x}|\theta)$ (i.e., it is proportional to the sampling density, with respect to the parameter vector). Multiplication of a likelihood function by a positive value that does not depend on the parameter vector leaves this proportionality requirement intact, so the result is another valid likelihood function. While we commonly refer to "the" likelihood function, this is actually a class of functions defined up to a positive multiplicative constant.