Do animals get sick from eating dirty food or eating off the floor?
I'm sure it varies wildly based on the animal and what they're eating.  In general, if in the course of an animal's natural feeding process it picks up a little dirt, it has evolved to cope with that.  Animal's behaviors and guts have evolved to fit their food source and lifestyle.  For a behavioral example, seals will eat rough rocks to help breakdown bones (and barf them up when they get too smooth).  Guts have evolved symbiotic relationships with microbes that are specifically adapted to the food source, termites have microbes to help them break down wood, whereas we don't have the microbe population to pull that off.  Interesting side note: it was found that one of the gut microbes in some Japanese people has gotten a horizontally transferred gene from a bacteria that lives on seaweed, probably helping them break down seaweed.

There could be a problem when the body is challenged with dirt that contains uncommon substances that the body isn't adapted to.  I'm sure if a cat rolled around in soil contaminated with a lot of arsenic it probably wouldn't be doing very well after licking off all the dirt.

This does bring up an interesting current medical hypothesis called the Hygiene Hypothesis.  Basically, these things you are talking about, utensils, antibacterial soap, etc. are all very very recent developments in human evolution.  As a result, proponents of this hypothesis think that our bodies are adapted to challenges by microbes and parasites, and depriving our bodies of that exposure when we are young totally screws up the development of our immune systems.  This would explain why we see a huge increase in allergies and autoimmune diseases in the developed world, our immune systems are totally miscalibrated, and freak out over harmless things like pollen (allergies), or worse, attack self-antigens (anti-insulin antibodies in type I diabetes).  Human children are animals, and baby animals should play in the dirt!