I personally wouldn't spend too much time worrying and second guessing what Templars are doing. There are never more then a few playing at once, and I imagine that they have staff eyes on them more than anyone else in the game. I imagine the staff will do the right thing and dock their status accordingly with the largely NPC population of their cohorts.
As for what a commoner thinks of it, a Templar stepping into the tavern and looking around would likely be pretty common, especially in a blue robe who is directly managing militia. I doubt it would raise any eyebrows and at worst cause people to do quick bows as they scoot around the Templar standing in the door.
On the other hand, a Templar sitting down at the bar and pounding a few brews should probably cause minor distress among commoners for two reasons.
First, Templars are scary. It is nice to have a Templar to kill or collar the evil magikers, keep out the gith, and drive away the Tuluki barbarian hordes. However, despite youru deep love and thankfulness for the Highlord's benevolence in providing Templars to save you from the horrible terrors of the wastes, Templars are still god damn scary. Magik, even "good" magik is scary stuff. Not only are they terrifying magik wielding avatars of your most benevolent and powerful God King, but they are Nobles on top of that. That means that they are pretty much a different species with inhuman motives living in an inhuman powers and spending much of their time in the inhuman world of nobility. Not many commoners can wrap their mind around what a Templar is and still have much thirst for drinking. Templars are comforts from afar and terrors up close.
Second, despite their awesome and terrible power and glory, commoners do have certain expectations of their Templars and Nobles. Commoners recognize these people as super human people with inhuman motives, but they still have expectations as to how these people act. The most obvious thing that a commoner expects is for a Templar (or noble), a high and mighty creature, to remain high and mighty. They don't need to be morally sound, incorruptible, or against snorting a line of spice. They do need to show themselves to be the better of commoners at all times, or else shatter the expectations of commoners.
So, don't worry if you see a Templar or noble slumming it. A staff member probably already knows and is applying the appropiate consequences. That said, you as a commoner are still able to react. You might not be in any position to make demands, but you can be disconcerted by a Templar who doesn't seem to be acting in line with your expectations of a Templar. I suggest worrying about what you should do IC, and let the staff worry about the consequences for the Templar ICly. Just because you don't see any consequences doesn't mean that there aren't any.