It depends on what layer the PC is infected and what component is infected by the malware.
If the malware lives in userspace or kernelspace on the operating system or is a bootkit installed into the bootsector of the operating systems primary hard drive, then it will be unable to interfere with any other operating systems that are booted on the system.
If it has infected firmware on a device then it may be able to perform malicious activities, depending on which bit of firmware and what the device can do will determine how compromised it is. For example, if the USB drive (e.g. BadUSB) or your CD/DVD reader device was infected, then it would be possible for that device to subvert the normal Tails boot process and inject malicious code during the boot process which would be very difficult to detect from inside the operating system and have complete control over it.
These lower level infection methods are problematic because they're hard to detect and they're hard to remove once detected. Research group LegbaCore created a proof of concept tool called "LightEater" which uses a remote exploit to gain code execution and then subvert the BIOS of the targeted system, once in the BIOS they are able to take total control of any operating system that is booted on the device, and invisibly operate it entirely outside of the view of the user, a demonstration of using it to steal cryptographic keys from Tails is available here.
You should also consider that creating a "livecd" from inside an infected operating system is a version of the "trusting trust" attack. If your operating system is untrustworthy, any media that you create with it could be subverted.
If a three letter agency was very interested in you and you were using a livecd with some read-only medium and you were exploited (by email, or whatever means) then they could potentially infect your computer, as mentioned if they attack the firmware of your livecd reader then they can make it read data that they control.
Also if you are running Tails from a livecd and you have your host operating systems boot drive plugged in then similarly they could infect it that way. There are ways to protect against this, by using secure boot with a TPM for example, but they're currently not very well supported or user friendly and still have loopholes that might allow attackers to subvert them.
Ultimately however, before you go down the paranoia rabbithole: it sounds like the malware you're describing, if it is indeed malware, is poorly written or is being operated by imbeciles (I.E. you detected because it was doing obvious things, it's not very well hidden, etc.). Unless you have reason to believe that intimidation is their intent then it seems unlikely to be a "three-letter agency" or any advanced attackers, it's probably only infected userspace or kernelspace, at worst a bootkit. A clean Tails install will protect you from all of those. I would still urge you to proceed with caution, if you do use Tails then never put in the Tails boot media while any other operating system is running (shut it down then insert the Tails media) and if possible create the Tails boot media from a trusted friend or colleague's computer system, or if that's not available then some random computer system that is available to you that would make it difficult to target you through.