1
This is a non-profit organization.
Sign 'non' (using the sign for 'not') followed by 'profit' to convey the concept of a non-profit. The forward motion of the thumb clearly negates the following concept.
2
The tickets are non-refundable.
Use the sign for 'non' before signing 'refund' to show that you cannot get your money back. Make sure your facial expression matches the negative statement with a slight headshake.
3
We need a non-toxic cleaner.
Sign 'non' and then 'toxic' or 'poison'. The sharp forward movement of the 'A' handshape emphasizes the absence of toxicity, making the meaning clear.