The answer is no. Using Wine does not make you immune to viruses. Any type of malware is code running on your computer and they can yes run on Wine.
There is a page on the Wine Wiki showing what can be done to protect yourself: http://wiki.winehq.org/SecuringWine
It should be borne in mind, however, that as viruses are mostly made for the Windows environment they will most likely not run as expected on Linux. A virus is programmed using various types of hacks based on a standard Windows installation (see this answer). When running in an environment like Wine, certain calls and checks will not work as expected and this will prevent the virus from working properly.
Unless, of course, the virus has been programmed to identify that it is running on Wine and from there do something. This is exactly why certain basic recommendations like not running Wine with root privileges should be followed. While running as root you would be leaving your system vulnerable for the virus to erase your files or even decrypt them, in order to charge for the description afterwards.
To give this answer I did some research and this is a recurring question in the community. My general perception is that although there is the possibility of a virus affecting Linux (by Wine), the probability is small.
About Antivirus, you can run Clamav.
Some questions:
https://askubuntu.com/questions/5031/what-if-i-run-a-virus-trojan-windows-exe-on-ubuntu-with-wine
https://askubuntu.com/questions/49255/risks-disadvantages-associated-with-wine
https://askubuntu.com/questions/562388/do-wine-viruses-only-work-while-wine-is-running
https://askubuntu.com/questions/361781/could-autorun-virus-affect-my-pc-via-wine