All Tcl commands need to originate from the same thread. Due to tkinter's dependence on Tcl, it's generally necessary to make all tkinter gui statements originate from the same thread. The problem occurs because mainWindow is instantiated in the tkinterGui thread, but -- because mainWindow is an attribute of tkinterGui -- is not destroyed until tkinterGui is destroyed in the main thread.

The problem can be avoided by not making mainWindow an attribute of tkinterGui -- i.e. changing self.mainWindow to mainWindow. This allows mainWindow to be destroyed when the run method ends in the tkinterGui thread. However, often you can avoid threads entirely by using mainWindow.after calls instead:

import time, threading
from tkinter import *
from tkinter import messagebox

def infinite_process():
    print("Infinite Loop")
    mainWindow.after(3000, infinite_process)


mainWindow = Tk()
mainWindow.geometry("200x200")
mainWindow.title("My GUI Title")
lbCommand = Label(mainWindow, text="Hello world", font=("Courier New", 16)).place(x=20, y=20)
mainWindow.after(3000, infinite_process)
mainWindow.mainloop()

If you want to define the GUI inside a class, you can still do so:

import time, threading
from tkinter import *
from tkinter import messagebox

class App(object):
    def __init__(self, master):
        master.geometry("200x200")
        master.title("My GUI Title")
        lbCommand = Label(master, text="Hello world", 
                          font=("Courier New", 16)).place(x=20, y=20)

def tkinterGui():  
    global finish
    mainWindow = Tk()
    app = App(mainWindow)
    mainWindow.mainloop()
    #When the GUI is closed we set finish to "True"
    finish = True

def InfiniteProcess():
    while not finish:
        print("Infinite Loop")
        time.sleep(3)

finish = False
GUI = threading.Thread(target=tkinterGui)
GUI.start()
Process = threading.Thread(target=InfiniteProcess)
Process.start()
GUI.join()
Process.join()

or even simpler, just use the main thread to run the GUI mainloop:

import time, threading
from tkinter import *
from tkinter import messagebox

class App(object):
    def __init__(self, master):
        master.geometry("200x200")
        master.title("My GUI Title")
        lbCommand = Label(master, text="Hello world", 
                          font=("Courier New", 16)).place(x=20, y=20)

def InfiniteProcess():
    while not finish:
        print("Infinite Loop")
        time.sleep(3)

finish = False
Process = threading.Thread(target=InfiniteProcess)
Process.start()

mainWindow = Tk()
app = App(mainWindow)
mainWindow.mainloop()
#When the GUI is closed we set finish to "True"
finish = True
Process.join()