Preheat the oven to 400 degrees F (200 degrees C).
Turn on a multi-functional pressure cooker (such as Instant Pot) and select Sauté function. Heat olive oil in the pot. Add onion, potato, carrot, tomato, celery, and garlic; cook and stir until slightly softened, 3 to 5 minutes. Sprinkle salt, cumin, cinnamon, and allspice over vegetables and stir until fragrant.
Pour in broth, water, and lentils. Close and lock the lid. Select high pressure according to the manufacturer's instructions; set the timer for 10 minutes. Allow 10 to 15 minutes for pressure to build.
Meanwhile, spread pita squares on a lined baking sheet. Spray with cooking spray and season with salt.
Bake in the preheated oven until toasted, about 8 minutes.
Release pressure carefully using the quick-release method according to the manufacturer's instructions, about 5 minutes. Unlock and remove the lid. Puree soup using an immersion blender. Stir in juice of 1 lemon.
Divide soup among bowls and scatter a handful of pita chips over each. Cut second lemon into wedges and serve alongside.