It depends on what version you have now. TurboTax for 2019 will support Windows 8 and higher, so if you are on Windows 8 you do not have to upgrade. It will not support Windows 7, so if you have Windows 7 then, yes, you should upgrade. You will have to upgrade before you can install TurboTax for 2019.
You should but for reasons well beyond Turbo Tax. Partway in to January 2020, Microsoft will cease all support for Windows 7. This means no service packs or security updates which would leave your Win7 device more vulnerable. I also suspect that the major Antivirus program makers will, at some point, stop releasing new products for Win 7. I have a 7+ year old Windows 7 PC which has served me very well but I will be getting a new Windows 10 PC before the end of the year. I also have a Windows 8.1 laptop and a Windows 10 laptop. Win 8.1 is a good operating system but Microsoft threw too many silly toys into it. I was reluctant to get a WIn10 laptop but I've been very pleased with it. For me Win 10 is more of logical jump from Win 7 than Win 8.1. It is now a solid operating system and doesn't some of those annoying "features" that Win 8 and 8.1 have.
TurboTax has announced a change regarding Windows 7 support. The TurboTax CD/Download software for 2019 will be able to be installed and run in Windows 7. However, they don't recommend it because of the security risks. The TurboTax CD/Download for 2020 will not be able to be installed in Windows 7. See the following FAQs.