Popular DAC that is known for its warm-ish sound?

North Star Supremo and Excelsio. The es9016 based DACs from NSD are more neutral but these two are very smooth and engaging. 

I cant believe nobody mentioned Chord. The Hugo series is excellent. 

Another nod can go to the AMR gear. I’d assume the new iFi Pro iDSD would have a similar tone but I haven’t heard it. 

Marantz gear is musical with just a tad bit of bite to it. I’d recommend the HD-DAC1 or the higher end SACD players. 

Any tube based dac should get there but then you might be chasing tubes instead of the dac. 

I haven't heard the TEAC 505 yet.  They can be found on eBay and Audiogon.  To be honest, given the prices available, I was going to order one without having heard it first.  A gamble for sure.  

Forsooth...

Audio Note shows a bunch of DACs on its website, but with no detailed information on any of them, including price. Why are so many models needed?

Audio Note's philosophy is to build high quality components at many different price points. Entry level shares many of the parts and design elements as higher level gear.

Pricing on DACs range from about $3000 to $100K, starting with the DAC-One up to the DAC5.
They are tube based NOS Dacs and present organic, realistic sonics. 
No affiliation, just a very happy owner. They can be found on the used market.

MHDT Havana or Paradisea+