One of my managers told me he had no time to actually keep an eye on what is going on in the team. He asked me to be his "eyes and ears". I think "snitches get stitches" and don't want to ruin my relationships with the colleagues. What should I do?
You'd be happy to accept his official promotion to Supervisor/Team Leader, of course!
Now, let's talk about the pay rise commensurate with this new position and attached responsibilities. And of course, your official title which will be obvious to your colleagues, who will fully understand the role and avoid any implication of spying.
Literally "spying" on your colleagues and running back to your boss with complaints and reports of misbehavior is sure to cause problems, much like you already suspect.
Instead, take it as an opportunity to be a leader. When you see a colleague doing something that you know should be done differently, talk to him/her yourself, and try to fix the problem without involving your manager. That is certainly in the spirit of your manager's instructions to you, because you're helping operations run smoothly. At the same time, you won't be considered a "snitch" because you're handling problems at a lower level without involving the manager.
Use the opportunity to bring process and policy issues to his attention, and make a point to discuss areas where the team as a whole could use more support from X department. Focus on areas for improvement, and give credit where it's due when the team or an individual does something worth recognition. Avoid mentioning interdepartmental conflicts or going the gossip route. Suggest that in addition to you bringing issues to his attention, a biweekly or monthly department meeting be scheduled so that other employees also have time to bring their concerns or suggestions to his attention with feedback from the entire group.
Really though -- a manager who doesn't have time to follow what's going on within his team? That's what managers are SUPPOSED to do. That's the primary job duty of someone in that role. If he's spread thin with meetings and other deliverables it's time to promote someone to a senior/supervisor role so that it's clear to the rest of the people on the team there is someone "officially" in charge of handling day-to-day operations, who's a known throughput to the manager.