Coulson's loyalty has always been to SHIELD, unwaveringly as far as we've ever seen (with the sole exception of his crisis of faith after learning about TAHITI).
That being said, there is one group who claims otherwise...
The "real" SHIELD
However, so far they haven't outright accused him of doing anything specifically disloyal, more that they simply don't trust him due to the effects of TAHITI and his subsequent actions.
A few examples:
- Coulson exhibited worrisome compulsions after his TAHITI treatments, manifesting most notably in an uncontrollable urge to draw symbols on any nearby surface. Notably, every other sufferer of this condition went homicidally insane.
- Coulson uncovered and opened a potentially devastating alien city, with (seemingly) little regard for the consequences beyond his need to know what was down there.
- Coulson has aided non-SHIELD elements (most notably Colonel Talbot) who have branded SHIELD a terrorist organization, as well as rogue elements (such as Raina) who pose clear threats to the world at large, and SHIELD specifically.
- Coulson has shown extreme loyalty to Skye, the only other person known to be connected to the TAHITI process, despite the clear danger she could pose to her team and the world at large.
...and most importantly...
- Coulson has become more and more calculating, dictatorial, and seemingly cold-hearted since his TAHITI treatments, and even more so since the fall of SHIELD. This has led him to maintain the policy of secret-keeping that Nick Fury established, and which that organization is ideologically opposed to.
The organization in question points to these facts as evidence that Coulson may have been altered, impaired, or otherwise changed from his TAHITI experiences, and as such may not be capable to lead SHIELD. The final point explains why they didn't simply contact him about it, and felt the need to work against him (or at least keep him under surveillance) without his knowledge.
Put simply, they're not doubting his personal intentions, so much as they're doubting whether it's a good idea to put a PTSD-suffering, obsessive-compulsive alien-blood-experimentee with trust issues and questionable allies in charge of a global intelligence and military organization.