Does Dusk Rose Lab's trigger go on the stack if you decide not to use it?

No, the trigger does not go on the chain (stack) if you decide not to use it.

According to the card text, Dusk Rose Lab reads: "At the start of your Beginning Phase, you may kill a unit you control here to draw 1."

Here is the breakdown of how this works:

1. Optional Trigger: Because the ability says "you may," it is an optional triggered ability.
2. Controller's Choice: Per RiftJudge FAQ #10373, you make the choice of whether or not to place the ability on the chain at the start of your Beginning Phase.
3. No Chain Involvement: If you decide not to use the ability, it simply does not trigger, and nothing is added to the chain.

If you do choose to use it, the ability becomes a pending chain item, but it will be removed from the chain during finalization if you are unable to pay the cost (killing a unit you control at that battlefield), as confirmed by the same FAQ.