I would say no, it wouldn't make much difference.
What people refer to as the cold harsh sound of digital is, in my opinion, the cold harsh sound of inferior converters. Coupled with the sound of inferior analog electronics ahead and after the AD/DA stages. Putting a tube amp ahead of poor converters is not really solving the problem. Putting the original signal through some great converters will give you a bigger, warmer, and non-harsh sound.
The thing about digital is that most people are recording and listening through poor-to-mediocre converters. So that, plus the way early DAT machines sounded, and you have this reputation for digital being cold and harsh. It needn't be that way any more, given the current availability of some really fine converters.
I'm referring to converters like those made by Benchmark, Mytek, Lavry, and on up.