Can I claim reservist travel to include rental car airline fees to travel to reserve duty weekends?

No, but you can claim the standard mileage rate if you traveled more than 100 miles and stayed overnight. 

"If you're a member of the National Guard or military reserve, you may be able to claim a deduction for unreimbursed travel expenses paid in connection with the performance of services as a reservist that reduces your adjusted gross income. This travel must be overnight and more than 100 miles from your home. Expenses must be ordinary and necessary. This deduction is limited to the regular federal per diem rate (for lodging, meals, and incidental expenses) and the standard mileage rate (for car expenses) plus any parking fees, ferry fees, and tolls." (IRS)

1. Login to your TurboTax Account 
2. Click on "Search" on the top right and type “2106”
3. Click on “Jump to 2106”