2006 Dodge Ram Upsizing tires

Yes, there is a problem with doing that. Not only will it effect the speedometer but the transmission relies on that info for shifting decisions and the ABS sensors will be reading wrong also. If you change tire or rim size, you should always try to maintain the same height to avoid these issues.

For the speedometer yes, but not the ABS.
The other issue here is that it will change the gear ratio substantially which will have a big effect on the fuel mileage and power.

The Service Manager or the Service Writer?
The service writer is not a tech. He's a service salesman.

Don't get me wrong, the problem is not critical but I assure you that if you have a transmission or suspension failure under warranty, they will be using it for an excuse to not cover you. I realize your truck is a 96 and that's why the guy was so quick to say "no problem". If it was still under warranty, he would have had a different answer.

There will be a power and mileage and shifting difference.

I misread the year of it. The general rule of thumb on ABS is that you can live with up to 10%change. Your just a little above that. I would be real concerned about your warranty though. They DO have the right to deny any warranty claim to do with transmission, differential or suspension unless that tire size is listed as optional for that vehicle in the owners manual. You'll probably never have an issue but it CAN happen. I don't see the reason or benefit to do something like that anyway.

That's right, it's not a giant amount but it's all height. By the sounds of it you have already done it so I wouldn't worry about it now. With the Hemi, you never notice the power difference.