How long would an original gameboy last on a modern lithium battery?
Let's compare the energy densities of the battery types:


Alkaline battery, 1.8 MJ/L
Lithium-ion, 2.36 MJ/L (best case)
Lithium battery, 4.32 MJ/L


Therefore, if a Gameboy lasted 10 hours on Alkaline batteries, in theory it would last 40.3 hours on Lithium (primary, non-rechargeable), or 20.4 hours on Lithium-ion batteries that occupied the same physical space.

If you used a phone battery, it depends on its size and capacity. Most phone batteries aren't as large as four AA batteries, so you would have to go by the capacity of the battery. Four AA batteries (in series for 6V) have a capacity of around 2500 mAh at 100 mA discharge rate. (The Gameboy Wikipedia entry indicates it requires about 0.7 watts which is a little more than 100 mA at 6V.)

An HTC One battery is a Lithium-polymer battery at 2300 mAh, which I'll assume is 3.7V LiFePO4.

Since the voltages are different, there would have to be some conversion which would introduce efficiency losses, but we'll assume 100% for now.

The four alkaline AA batteries then are 2500 mAh * 6V = 15000 mWh.

The HTC One battery is 2300 mAh * 3.7V = 8510 mWh.

Based on this, you'd get roughly half of the runtime using your phone battery.