Yes, you can include in medical expenses the costs of buying, training, and maintaining a guide dog or other service animal to assist a visually-impaired or hearing-impaired person, or a person with other physical disabilities, including emotional Support Animals expenses.

The costs of buying, training, and maintaining a service animal to assist an individual with mental disabilities may qualify as medical care if the taxpayer can establish that the taxpayer is using the service animal primarily for medical care to alleviate a mental defect or illness and that the taxpayer would not have paid the expenses but for the disease or illness.

Where to enter medical expenses in TurboTax:  

While inside the software and working on your return, type medical expenses, sch a in the Search at the top of the screen (you may see a magnifying glass there).  There will be a popup that says Jump to medical expenses, sch A.  Select that to get to the general area. 

Please see the following link for a comprehensive list of deductible medical expenses.  IRS Medical Expenses

An Emotional Support Animal (ESA) is NOT the same as a Service Animal. The ADA Laws only cover a Service Animal, they do not carry over to an Emotional Support Animal. Please understand that to claim your Emotional Support Animal as a Service Animal is in violation of the ADA Laws. A Service Animal provides a service to a disabled individual, i.e. leading a blind person, picking up a dropped cell phone. If your ESA provides support to you while at home, or public transportation, that does not qualify them as a Service Animal. ESA's are not provided the same public access that Service Animals are.

@Dog Lover is correct per ADA laws "Service animals are defined as dogs that are individually trained to do work or perform tasks for people with disabilities." which does not include emotional support animals. Please take a look at https://www.ada.gov/regs2010/service_animal_qa.html

I have a military working dog that provides a daily service at my airport for explosives detection. This dog is certified for his services by DHS and highly trained. We both work as federal employees but only get partial compensation for some of the expenses of maintaining a working dog. My question is, can i claim some of the out of pocket expenses i make to keep my dog fit for duty since my employer will only cover to an extent? Thank you.

Unfortunately not at this time.  The Federal Tax Cuts and Jobs Act that took affect on 1 January 2018 eliminated the "miscellaneous itemized deductions" category which included un-reimbursed employee expenses for tax years 2018-2025.  

There is no such thing as an "emotional service dog." You're thinking of an ESA which is a pet with a letter from a doctor saying you need the animal for emotional support. In no way is the animal task trained for a disability. Because of this, no, you cannot write off any expenses for your ESA.

This Department of the Treasury letter asserts that “the costs of buying, training, and maintaining a service animal to assist an individual with mental disabilities may qualify as medical care” and may be deductible as a medical expense. 

However, the letter makes it clear that the question of fact would be “determined on a case-by-case basis”. 

Certainly, the burden of proof would be on the taxpayer claiming the deduction.

See also here.

Question - I have a Certified Therapy Dog, who has worked with disabled people in our local hospital system (including the Psych ward). He also worked in Assisted Living Facilities with Alzheimer's patients. Following this, he worked in our local high school - primarily with kids who had mental disabilities, physical disabilities, or social issues. He worked for 10 years and his work is documented by the organization.  Couldn't my expenses for his required medical vaccines, annual certification, and my personal gas mileage be written off? And, if so, where do I make this entry in TurboTax?

Question - I have a Certified Therapy Dog, who has worked with disabled people in our local hospital system (including the Psych ward). He also worked in Assisted Living Facilities with Alzheimer's patients. Following this, he worked in our local high school - primarily with kids who had mental disabilities, physical disabilities, or social issues. He worked for 10 years and his work is documented by the organization.  Couldn't my expenses for his required medical vaccines, annual certification, and my personal gas mileage be written off? And, if so, where do I make this entry in TurboTax?

Yes, you can deduct at least some of the expenses for your Certified Therapy Dog.  Here is a discussion pertaining to this topic. There are also other sources out there that support that if the dog is going to nonprofit hospitals and the like that this is a charitable donation expense.  Good recordkeeping is key here. The mileage to get to the visits would be a charitable deduction. 

The deductions for your situation would be charitable, not medical deductions.  They would apply if you are itemizing deductions and not taking the standard deduction.  

Charitable Donations