How to run multiple while loops at a time in Python [duplicate]

No, that's not the way it goes. You have an application loop so use it. time.sleep, pygame.time.wait() or pygame.time.delay is not the way to wait or delay something in a typical application. The game does not respond while you wait. Use pygame.time.get_ticks() to measure the time.

In pygame the system time can be obtained by calling pygame.time.get_ticks(), which returns the number of milliseconds since pygame.init() was called. See pygame.time module.

Calculate the point in time when the text needs to be changed. Get a new random text and render the text Surface after the current time is greater than the calculated point of time. Calculate a new random point in time which is greater than the current time:

next_render_time = 0

while run:
    current_time = pygame.time.get_ticks()

    # [...]

    if current_time >= next_render_time:
        currentFace = random.choice(face)
        faceDisplay = myFont.render(str(currentFace), 1, (0,255,0))
        next_render_time = current_time + randint(5, 10) * 1000

    screen.fill((0,0,0))
    screen.blit(faceDisplay, text_rect)
    pygame.display.flip()

The time is multiplied by 1000 (randint(5, 10) * 1000), since the time unit of pygame.time.get_ticks() is milliseconds.

Actually, you can also use a timer event. See How do I use a PyGame timer event?.
In your case, however, the time interval is not constant. It's a bit odd to use a timer event on a dynamically changing interval. I would prefer to use a timer event for an action that needs to be performed at constant intervals. But that's just my opinion.

Complete example:

from random import randint
from time import sleep
import pygame
import pygame.freetype
import time
import random
run = True
pygame.init()

#faces
face = ['^-^', '^v^', '◠◡◠', "'v'", '⁀◡⁀']
talkingFace = ['^o^', '^▽^', '◠▽◠', "'▽'", '⁀ᗢ⁀']
currentFace = random.choice(face)

#background
screen = pygame.display.set_mode((800,600))

#font and size
myFont = pygame.font.Font('unifont.ttf', 100)

#face render
faceDisplay = myFont.render(str(currentFace), 1, (0,255,0))

#center and draw face
text_rect = faceDisplay.get_rect(center=(800/2, 600/2))

next_render_time = 0

#prevent crashes
while run:
    current_time = pygame.time.get_ticks()
    for e in pygame.event.get():
        if e.type == pygame.QUIT:
            run = False
 
    if current_time >= next_render_time:
        currentFace = random.choice(face)
        faceDisplay = myFont.render(str(currentFace), 1, (0,255,0))
        next_render_time = current_time + randint(5, 10) * 1000

    screen.fill((0,0,0))
    screen.blit(faceDisplay, text_rect)
    pygame.display.flip()