The OTP part of the question is a distraction from the main point: it depends heavily on hypothesis on the plaintext if learning a sub-string of it helps towards finding the rest of the plaintext.
For plaintext consisting of independent symbols, no, learning a substring does not help towards guessing the rest. Argument: use of the OTP implies that we can't learn anything from the ciphertext. That leaves the given plaintext substring as the only information about the plaintext. Because symbols are independent, revealing some is of no help to guess the others.
For other type of plaintext, including as in practice and as considered in standard theoretical hypothesis, yes it can help, possibly heavily. Illustration: for plaintext known to be a 14-symbols English word, learning that it starts with
HYPOTHE helps a lot towards guessing that the rest is
Note: if the question was to be read as: does knowledge of the OTP ciphertext help to guess the end of the message knowing its start?, the answer is no, under the assumption that the pad was uniformly random and not reused.