How does 1099 pay affect unemployment?

Yeah, you will likely lose the unemployment.  Of course, if you turn down a job, you also lose it.

When you certify for your weekly unemployment benefits, you must answer the questions HONESTLY.   That means you must report any self-employment income.   Yes, it will probably affect how much you collect. 

"No taxes taken out" means YOU will be paying all the taxes.  You will owe 15.3% self-employment tax on all earnings over $400 in a calendar year PLUS whatever income tax you will owe.  

The IRS has criteria by which you can determine whether you actually qualify as a self-employed person or whether the guy is just trying to shift the tax burden onto you instead of paying his share of employment taxes including worker's comp and unemployment insurance.

If you thought you could earn $600 week, pay no taxes and keep collecting unemployment, you'd be breaking multiple laws AND you'd be caught because your 1099 income is going to be reported to the IRS by the guy who pays you.

First, does the job meet the requirement of a 1099.

If not, don't touch it.

Also, being an independent contractor means you pay more taxes and the company that hired you pays less. 

- So again, if doesn't meet the requirement of a independent contractor = don't touch it.

If you take the job, you need to continue to complete your claim from each week.  You have to continue looking for permanent work.  You have to claim this money.  If the $800 is including the extra $300 each week, then you probably will not get any additional UI money if you are earning $600.