Minecraft auto-backup utility
import time
import os
import sys

#class for parsing settings.ini
class parseIni(object):


Python style guide recommends names like ParseIni

    def __init__(self, fileStream):


Python style guide recommends argument like file_stream.

        self.lines = fileStream.readlines()


You don't use the lines again, so why are you storing them on the object. There is not a whole lot of point in calling deadlines, because you can actually just pass the file object in the for loop for the same effect

    self.options = []
        for line in self.lines:
            if "=" not in line:
                continue


I suggest almost always avoiding continue. As it is continue does nothing, replacing with pass will have the same effect. The whole if can be simplified by inverting the condition

            else:
                self.options.append(tuple(line.replace("\n", "").split("=")))


For clearing out excess whitespace, I recommend using line.strip(). I'd also suggest having options be a dictionary. Then you could do

 key, value = line.strip().split('=')
 self.options[key] = value


It'll make lookup more efficient and is a bit simpler

    def getValueOf(self, option):


Awkward and longish name. I'd have this be __getitem__, and then you can access it with the [] operator

        for op, value in self.options:
                if op==option:
                    return value


This is the for-if anti-pattern. Whenever you find yourself writing an if which should only be true for one instance of the surrounding loop, you should look for a better solution.

Also, what happens if the option is missing? You should probably raise an exception.

#Tries to open the settings file, if unsuccessful then exit program
settingsFileStream = None


No point in doing this. 

try:
    settingsFileStream = open("settings.ini")
except IOError:
    print("ERROR: SETTINGS FILE NOT FOUND.", file=sys.stderr)
    raw_input("Press &lt;ENTER&gt; to exit....")
    sys.exit(0)


You should probably print the exception itself. As it is you are throwing away potentially useful error information.

#Initialize parser
settings = parseIni(settingsFileStream)

#Precondition: settings file was opened successfully
#Assigning values to required variables
timeDelayInMins = eval(settings.getValueOf("timeDelay")) #For user information


Using eval is a bad idea. It'll try to evaluate any valid python you pass to it which could allow an external file to cause your application to do bad things. You also can't be sure what kind of data you got back. Instead pass the incoming string to int or float to safely convert.    

timeDelay = timeDelayInMins * 60 #Converting to seconds, for time.sleep() method
sourcePath = settings.getValueOf("sourcePath")
destinationPath = settings.getValueOf("destinationPath")

#Runs the game
os.system("Minecraft.exe")

#Prints out Welcome message
print()
print("|--------------------------------------------------------|")
print("|WELCOME TO MINECRAFT AUTO BACKUP UTILITY!               |")
print("|--------------------------------------------------------|")
print("|REFER TO THE settings.ini FILE TO EDIT SETTINGS FOR THIS|")
print("|PROGRAM. e.g. TIME DELAY FOR AUTO BACKUP, PATHS etc     |")
print("|--------------------------------------------------------|")
print("|TIME DELAY FOR AUTO BACKUP IS {:0.2f} MIN(S)        |".format(timeDelayInMins))
print("|--------------------------------------------------------|")
print()

#Variable to keep tract of backups done per session
backupsDone = 0

#mainLoop runs forever, backups "saves" folder every after ever timeDelay mins
while True:
    time.sleep(timeDelay)
    os.system('robocopy "{}" "{}" /e'.format(sourcePath, destinationPath))


Python has function for copying files. Consider using that instead of calling to another program.

    backupsDone += 1
    print
    print
    print("|---------------------------------|")
    print("|{} BACKUP(S) DONE THIS SESSION.  |".format(backupsDone))
    print("|---------------------------------|")


It would also be a good idea to move all this logic into a main() function.