I'm not sure exactly what you are referring to with the term "non-qualified IRA", but it seems to be reported as a normal distribution from an IRA. As such, when you roll it over to a similar IRA, you can designate it as such in TurboTax after you enter the 1099-R form. You will see a screen that says What Did You Do With The Money From (name of investment)? You will answer I moved this money to another retirement account and I rolled over all of this money to another traditional IRA or other retirement account. If you do that, it won't show as taxable on your tax return.
However, if the original transfer was from a qualified traditional IRA account in which the investment consisted of pre-tax contributions, the transfer to a non-qualified IRA would likely have been taxable. Sometimes such a trustee to trustee transfer won't be reported on a form 1099-R, but normally that would only occur if it was from a qualified plan to another qualified plan.
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