I'm really not sure Simadsen. I seem to remember this issue of slight residual sound has been brought up before and that the results of that query were that the amp could not ever be made completely quiet but I think the circumstances may have been different.
On a technical note for my fellow geeks out there, it seems to me that some of this could be a consequence of some of the internal circuitry being referenced to a decoupled DC voltage rather than to ground, hmmm, that would most likely make it frequency dependent so that is not likely to be the problem. Or it could be a factor in the digitally controllable Digipots' construction. Looking into the Digipots they use in the H&Ks I found that Analog Devices says: "The minimum wiper resistance is at the wiper’s first connection at the B terminal for zero scale. This B terminal connection has a minimum wiper contact resistance, RW, of typically 70Ω." My guess would be that is where our small residual noise may be coming from.
So ok, where is our solution? I don't think there is anything you can do in the amp. One way would depend on whether you use your Fx Loop or not. Assuming your tuner has isolation when in use and the infamous "true bypass" when not, have you thought of putting your tuner in the Fx Loop and turning that on as a way of switching to tune? Even if you are using the Fx Loop, as long as your tuner has true bypass it shouldn't have any effect at all on the signal when you aren't using it. Remember that the signal at this point is somewhere around line level not at guitar input level.
To everyone out there, don't just swallow the common knowledge that says it must have a deleterious effect on your tone just because it is there! People make this mistake. They put their guitar into a pedal via a cable then into the amp via a second cable. That involves two lengths of cable and that is double the capacitance. Capacitance on a guitar output doesn't just "draw off treble", it completely recalculates the pickup/load resonance and that can be very obvious as more than just a loss of top end. Load capacitance on a guitar is the biggest source of "tonal change", (I just can't being myself to type that bloody common usage phrase once again). When the pedal is on it acts as a buffer isolating the guitar from the second length of cable. The guitar is loaded sensibly so it sounds good. When the pedal is off so the true bypass is in action, the guitar sees both the lengths of cable and sounds a touch flat. Ergo in most peoples' thinking, the pedal must be causing it to suffer from "sucked tone", (Aaarghhh, that damned phrase!
). Wrong! The true bypass is doing what it should, passing the signal through exactly as it was. The problem is the guitarist's lack of understanding of how his guitar is being loaded. It is totally predictable and completely preventable. The answer is to NOT use true bypass but to make sure there is a buffer of some sort at the point of the pedals so that even with all pedals off the guitar is still isolated from the second cable. This could be in a Behringer style Off mode where the input buffer of a lot of their earlier pedals was left in circuit to combat this effect. That way the guitar sees the same capacitance from the guitar/pedal cable to its input buffer stage and that stays constant whether the signal after the buffer stage is passed on to the pedal circuitry or straight on to the amp. And Behringer and others get slated by the non-technical for having that in their pedals.
Try it in and out of the signal path and listen - honestly. Set it up like this. Set up your amp without any effects or pedals. Put the tuner in the Fx Loop and turn it off, i.e. onto bypass. Have someone switch the Fx Loop on and off at random, they toss a coin to decide which, on the amp while you are unaware of what they are selecting. Play and listen and see if you can spot any differences when you don't know whether the tuner is in circuit or not. Choose whether you think it is currently set with the tuner in or not. Your friend should mark down both the setting, in or out, and your guess and not tell you of any results until you have completed a good number. Do this a good number of times then see how many correct guesses you get. I would suggest the more you do it the closer to 50% right you will get. The contacts in the Fx Loop switching will have as much effect as the contacts in the tuner and we don't even worry about that!
If the Fx Loop is out because you rely on it already and you are still fussy about going through the tuner when it isn't in use then I really think you may be stumped.