You can deduct your out-of-pocket costs for "boarding" the dog (e.g. food, vet bills, supplemental medications, bedding, etc.) but only if your employer does not reimburse you, you itemize deductions and your total unreimbursed employee business expenses exceed 2% of your adjusted gross income (AGI).
These would be "Job-Related Expenses", entered in the Employment Expenses interview of Federal Taxes > Deductions and Credits. You can "Jump to" this interview by entering "2016" in the Search Box (upper right of the TurboTax screen) and selecting "Jump to 2016"