The output voltage of the charger board is marked with BAT. You should connect it to the Vin pin of the Teensy, which then will convert the batteries 3.7-4.2V down to 3.3V.
Of course you need to connect the grounds of the charger board and the Teensy (on the charger board ground is marked with G).
The 5V pin on the charger board is no output. The board ...
This problem is usual if you are in shortage of RAM.
Adafruit's libraries wastes lots of memory. You should be very careful to run their stuff if you are in shortage of memory.
I suggest you to try another library which quite old, but uses much lesser memory to read DHT sensor’s data: