For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
As with so many things involved in creating a memorable brand, the answer is "it depends". In this case, it really comes down to establishing whether or not there is a fundamental element of your brand position that requires or suggests multiple color schemes. Otilia's example , FedEx, is a good one. The core logo was reimagined in different colors to support the company's diversification of services. In other words, it became a cohesive system, with clear rules for when to use each version. This was, of course, relatively easy for FedEx to pull off, as their visual identity was well established. I've seen this done well, many times for other companies, often represent different divisions, or even different customer experiences.
Your example, on the other hand seems to suggest that you want to introduce variation primarily for aesthetic reasons. Unless your logo is very well known, I'd think twice...it will, as others have said, hurt the recognition and recall of your mark. Good luck!