Technically yes — sound will come out. But a phone typically delivers 1–5 mW into 300Ω, which means you will run out of volume before you run out of quality. Use the calculator above to see exactly how much power your headphones need.
Beyond a basic quality threshold, rarely. Modern DAC chips are extremely good. The audible differences between a $100 DAC and a $1,000 DAC, in a controlled blind test, are smaller than most people expect. Amplifier quality and headphone choice have a larger impact on what you actually hear.
For critical listening: Beyerdynamic DT 770 Pro (closed) or DT 990 Pro (open). For casual listening: Audio-Technica ATH-M50X. All three are used daily in professional environments. None of them require you to spend more to enjoy them.
Tube amplifiers add measurable second-harmonic distortion. Many people find this pleasant — it is warm, smooth, and forgiving of poorly recorded music. If you are trying to hear recordings accurately, a solid-state amp is more appropriate. If you want music to sound a certain way regardless of the recording, a tube amp may suit you.
It is an engineering achievement and sounds excellent. Whether the gap between it and a $500 studio DAC justifies the price difference is a question only your bank account and ears can answer together. My honest answer, having owned both: no.