For example, your app icon would be blue and black and your logo on printed shirts would be red and black.
This is absolutely fine as long as you create and incorporate it into your branding guide and keep it consistent.
You should define theses alternate logos, and then creates rules around how they are used and when it's appropriate.
But above all, what is the REASON you want to use different colors? If it's "just because" then no, you shouldn't.
Keep in mind, the branding of your logo is about being recognized and creating an emotional connection with your customers that makes them feel they trust your product and your message.
Changing your logo constantly means your customers no longer recognize you as easily.