[mikewill!]

I did a DNA worth about $600 that has already been refunded. However there is an ongoing investigation with UPS. A delivery driver left a note on my door asking whether I received the package or not and asking for my signature. The note also says that the driver will come back tomorrow to get the note and try to speak with me.
I'm thinking I could either write on the note or tell the driver pretending to be a family member, saying the person the package was meant for has moved. I don't want to ignore it because the driver could eventually come and talk to a family member that might reveal my identity.

[HeadxHunter]

So they left your package unattended and without a signature, and it got stolen? You're being asked to sign a 'denial of receipt form' which is not typically legally binding, so sign them so they can process claim. Don't dig a hole deeper then it needs to be.

[mikewill!]

So they left your package unattended and without a signature, and it got stolen? You're being asked to sign a 'denial of receipt form' which is not typically legally binding, so sign them so they can process claim. Don't dig a hole deeper then it needs to be.

There's a picture of the package delivered. Also it says that this can be followed up by a police report.

[InternetRockStar]

So they left your package unattended and without a signature, and it got stolen? You're being asked to sign a 'denial of receipt form' which is not typically legally binding, so sign them so they can process claim. Don't dig a hole deeper then it needs to be.

There's a picture of the package delivered. Also it says that this can be followed up by a police report.

Ignore completely

[Elsdebo]

So they left your package unattended and without a signature, and it got stolen? You're being asked to sign a 'denial of receipt form' which is not typically legally binding, so sign them so they can process claim. Don't dig a hole deeper then it needs to be.

There's a picture of the package delivered. Also it says that this can be followed up by a police report.

That is likely just a scare tactic unless your SE was very very obvious.