The application which you "install" to use the VPN service is local code. If that code is hostile, then you have lost: as soon as you run malicious code on your own machine, that malicious code can more or less hijack your whole machine.
Theoretically, if you run the code as a non-privileged user, and your operating system ensures perfect isolation against non-privileged local users, then you might recover from running malicious code without reformatting your complete hard drive. However, this is not a reasonable foundation, because no practical OS succeeded at enforcing such an isolation. All OS have local privilege escalation holes. Moreover, a VPN must hook itself in the OS network subsystem, which will require some non-trivial privileges, so the point is moot.
Assuming that the TunnelBear's application is not malicious, then the VPN, being a VPN, protects data only in transit, between your machine and the VPN exit point, on TunnelBear's servers. These servers, by construction, see all your data unencrypted. A VPN (when done properly) is like an armoured steel tube between your machine and the exit point; it does nothing whatsoever for traffic beyond that exit point.
If you want to protect your data against inspection by the VPN maintainers, then you must use end-to-end security with whatever machine you are trying to contact, which basically means SSL (i.e. HTTPS). This begs the question of why you would want a VPN in the first place, of course.