Are dimensionless physical constants predicted to be rational, irrational, or transcendental numbers?

It depends on the constant you are talking about. For instance, it is an observational fact that the dimensionless constants $q_1 / q_2$, where $q_1$ and $q_2$ are the electric charges of two arbitrary particles, are rational numbers. It is a longstanding question of theoretical physics to understand why it is so.

In some cases, one can explain why constants are indeed rational. For instance, if you are talking about charges with respect to a non-commutative gauge group, then the general theory of representations of Lie algebras shows that ratios of charges have to be rational. One can not directly apply this argument to the problem of the previous paragraph, because the gauge group of electromagnetism is $U(1)$, which is commutative (we also say "abelian").

On the other hand, to my knowledge nobody knows anything about ratios like $m_p/m_e$, where $m_p$ is the mass of the proton and $m_e$ is the mass of the electron. From mearurements we know that this can't be a rational number with a small denominator, but nothing more.

A last remark : I'm not sure many physical constants (if any) have been computed "from theory", if by "theory" you mean "string theory", as your tag may imply.