How long does a DAC take to burn in?

The general thought is 100 hours.  However, I have seen certain components and metals that may take well over 200 hours to burn in.  Rhodium takes an extremely long time to burn in and also certain electrolytic capacitors take long hours.  It entirely depends on what components your equipment uses and your definition of what "burned in" sounds like.  Some people have higher thresholds than others.

There is no benchmark, it varies by component. In detail, what I have personally experienced is a very wide break-in period variance with no empirical explanation other than an anecdotal summation that the better the build quality, the longer the break-in period. Again, this is only my personal observation .... your experiences may indeed be substantially different.

For example,

(1) My HT electronics (Bluray spinner, AV kit, ~ $6000) break-in period took about ~ 100 hours +/- break-in time, with arguably and ill-defined nominal changes after that initial break-in period.

(2) In contrast, my 2-channel "A" system electronics (digital source and integrated amp ~$25, 000) with a much higher build quality and resulting pricepoint strata, took at least near- 400 hours initial break-in time to reach a certain plateau where any further changes curves now flattened out.

There were some more subtle further slower evolution changes over the next few hundred hours of use.

Note: The break-in period audio performance improvement curve was not a pure straight-line linear one , but rather a series of incremental non-predictable ad-hoc steps.

Without prejudice to any the above , the new audio performance that appears "warmer", and "cleaner" as you highlighted , may actually be a result of the following rather than any enhanced break-in period:

(a) it has a markedly different sonic signature of the new gear brand itself, that you personally (emphasis added) prefer, in conjunction with
(b) a more pleasing audio performance in your bespoke system ( again, emphasis added...) provided by a a "better" system synergy now overall. 

How do you know when it’s broken in? Things change in 100 hours of playing music. Either intentionally or unintentionally so how can you keep track of the sound?

My Benchmark DAC1 was bright sounding at first, then a little bit muffled and then open, smooth and more extended.  It took about 100 hours at least to this point.  There were still very small changes after that, I'm sure.  My class D amp went to about the same changes but it was very slow - about 4x that (400 hrs).