An employer in Ohio has implemented mandatory shutdowns for employees. They are requiring the employees to either use vacation time for these shutdowns or not get paid. As a salary exempt employee, is this company required to pay me for this mandated time off (I have a letter from the company from my pay raise for the year stating what my annual salary will be)? Can they mandate salary exempt employees to take this time off? Thanks in advance for your help.

This is an oversimplification, but for purposes of your question basically there are two types of law:
- Labor law is law the government imposes on employers. The main federal labor law is called FLSA (Fair Labor Standards Act). This is the law that defines things like Exempt and Overtime. Under that law there are some restrictions on docking salaried exempt employees. For the sort of thing you are talking about under labor law, your employer can arrange to have entire workweeks involuntarily not worked only without pay, but not workdays involuntarily not worked. Past that vacation/PTO is not a federal law concept. FLSA is concerned with pay based on time worked, not vacation/PTO balances. As long as the correct amount is paid, the feds do not care about what happens to the vacation/PTO balance.
http://www.dol.gov/dol/allcfr/ESA/Title_29/Part_541/29CFR541.602.htm
- The other type of law is Contract Law. These are obligations created between parties that are not imposed by the government. In theory your offer letter could maybe possibly rise to the level of an enforceable contract, although I have never seen an offer letter for which this was true. Still, anything is possible and you can always try showing yours to a local attorney. The key to any Contract Law situation is that a local attorney has to read the actual contract. These things tend to be very exact wording specific and very state law specific.

Do holidays count as time worked? We are being require to take December 22, 23, and 26 off. We will be paid for December 24 and 25 (holiday pay) of that week. Is the company responsible for paying us on the 22, 23, and 26? This would be for salary exempt employees.

Maybe. The Exempt Salaried rule (29 CFR 541.602) states that (among other things) the employer can legally dock an Exempt Salaried employee anytime an entire workweek is not worked. You have listed what seems to be a calendar week. A workweek is any fixed 168 hour period of time chosen by the employer. It might look like a calendar week. It might not.

Do you know what your employer's workweek is?

If you are asking whether or not holiday pay counts as time worked and therefore the entire week must then be paid, no, it does not.