No, the Runge phenomenon is known not to affect Shamir's scheme.
Remember, the point of Shamir's scheme is not actually to form an approximation over an interval; instead, it's to encode a secret in a randomly chosen polynomial, and then divide up clues to that polynomial so that, with enough clues (shares), someone can reconstruct the entire polynomial (and hence recover the secret).
(Now, in practice, we don't bother reconstructing the entire polynomial; instead, we just recover the coefficient that is the secret. However, that's just a detail; we could, if we wanted to, recover the entire polynomial).
So, in terms of accuracy, the question is "do we recover the polynomial with sufficient accuracy so that we can uniquely identify the secret". And, the answer to that is, yes, we can (because computation in Finite Fields are done precisely, with no rounding errors).