When to use investigate, and when investigate into?
I agree with the comments from Weather Vane and Peter Jennings - in most cases "investigate" implies "into", and out of context "investigate into" looks like sloppy shorthand for "carry out an investigation into".
More information on the context would have been useful. For example "The doorman allowed the detectives who had been assigned to investigate into the building." would be correct English - though commas, parentheses or dashes around "who had been assigned to investigate" would improve clarity.