For a lack of other schematics I will use the schematics of the Pi 1 B, Rev 2.1 here. Unfortunately the description of the Broadcom chip itself lists no electrical spec (BCM2835).
Let's have a look where the +5V (coming from the powering USB port and passing through the poly fuse) are used on a Pi:
- the main 3V3 switching voltage regulator (an NCP1117)
- +5V0_HDMI pin at the hdmi connector (including the CEC lines)
- the down-stream USB ports
- the VDD_BATx pins of the BCM2835
So assuming USB and HDMI are not going to be used on this application we still have two points to consider.
- NCP1117 regulator:
It's a low-dropout positive, fixed voltage, linear (non-switching) regulator. At 3V3 output and an current of 500 to 800 mA a dropout voltage of 1.0 to 1.1 V will occur. So it is not suited to provide clean 3V3 from a single cell lithium ion battery.
Bonus "Will 3.7V harm the RPi's 3.3V input?" - I do not know yet, but I'd guess so...
So the save way would be a switching regulator (probably a buck-boost DC/DC converter) to provide the 3.3V, e.g. here.
- The VDD_BATx pins
These are tied to the BCM's internal SMPS (switching-mode power supply) to create the "core voltage". You certainly want your main processor's core voltage clean. The use of the BCM2835 in mobile applications at least strongly suggests that this converter will operate safely from a single cell lithium ion battery. This wiki entry documents the core voltage to be 1.2V and it is not unreasonable to expect the internal SMPS to generate 1.2V safely from 3.3V.