Will my TSP investments have an impact on my eligibility to deduct IRA contributions?

They might. Investments in a traditional TSP balance reduce your modified adjusted gross income; depending on your income, that might bring that level down to the point where an IRA contribution becomes deductible—or more of it becomes deductible if you are in the deductibility phase-out range. For the pertinent levels, see www.irs.gov/Retirement-Plans/IRA-Deduction-Limits.