Are there any tube amps that don't give off any---

Yes-

As others have mentioned - Carver Crimson 275 -

https://www.bobcarvercorp.com/copy-of-crimson-raven

@austinstereo.  That is interesting. I am not concerned about the issue of injury, as I never let folk get near the tubes. Your client has a point though if he had little children or pets that could get injured!
The issue I brought up in my OP, has to do with the fact that the room size is also crucial. The smaller the room, the bigger the problem.
Some folks probably don't comprehend why this would be a problem, as they enjoy a large enough listening  room that a warm/hot amp ( heat wise) won't ever be an issue. 

You could consider a class D amp with a tube input buffer. The Purifi modules add very little of their own sound signature, but require a front-end driver board to provide some gain and impedance matching. There are a couple of companies that make amps using these modules (which are very efficient) that use a triode-based front end.

https://vtvamplifier.com/product/vtv-amplifier-stereo-purifi-audio-1et400a-amplifier-with-vtv-vacuum...

ATM makes an Korg NuTube based input buffer. I assume they make a complete amp that uses this buffer, but I don't know the model. The ATM buffer seems a bit pricey for what it is, but it might sound good. 

https://atm-audio.com/product/buffer-korg-nutube-for-hypex-nc500-and-purifi-audio-1et400-v2/

I haven't heard either of these, but have heard an amp using the Purifi module and it is very clean and detailed.

@jaytor Isn't the Purifi module a Class D module? I'm not really interested in a Class D amp, mainly due to the fact that all of the Class D amps i have heard are too sterile sounding to my ears. Even the 'best of class' Jeff Rowland models.

@daveyf  The amount of heat generated by a particular amplifier is predictable to a close approximation based on its current draw.  All of the current drawn by an amplifier is either converted to sound, which is subsequently converted to heat, or else it is lost immediately to radiant heat.  This is a consequence of the 1st and 2nd laws of thermodynamics.   There is a small amount of energy (either as sound or as heat) that escapes the room, so the amount of heat retained in a room is somewhat less than the amount of current the amplifier draws. 

This holds regardless of whether you are talking about tube or solid state.  It is just the laws of physics.  Any energy introduced into the room that is not lost through the walls will be retained in that room and sooner or later be converted to heat. 

 If you want to solve the problem without relying on AC to remove heat, the best bet (especially if you want class A amps) is to build a system around high efficiency speakers.  These usually can be driven by amplifiers that draw relatively little current even though they are class A.  The other alternative is to use a Class AB amp, which will draw less current when the music doesn't require it. 

As for me, I just put in a really nice quiet two stage AC system in my listening room, which had formerly been so hot I couldn't use my favorite amps during the summer months.  My new AC cost 6K, which was a pittance compared to my investment in audio.   It is a high efficiency system that will pay for itself easily over its lifetime.