Of course you don't have to pay them - you just might not like the result.
As a matter of law - given that I am not a lawyer - I am not aware of any requirement for a company to pay employees business-related expenses. An example might be having a cell phone, and according to this article companies aren't required to pay for you to have a cell phone even if they require you have one and use it as part of your employment.
The primary areas where law does exist relates to company uniforms with a logo (in a very limited number of US states) and necessary personal safety equipment (in California and maybe only few other states). All other tool requirements for a job are not prohibited by law, so long as they are not illegally discriminatory (such as requiring people of a certain race or sex to buy something but no one else, etc).
So a company can require all sorts of things, from having an internet connection to cell phone to laptop to specialty tools and equipment of all sorts, and they are even allowed to deduct the cost of some things from your pay - just so long as you still get paid minimum wage after the deductions.
With all that said, the company's previous payments of fees and willingness to pay a monthly internet fee does not obligate them to pay other fees too, such as moving/installation/etc. They may even decide to no longer provide internet service at their expense and just require you to provide it as a condition of employment.
You can insist on it with your employer, and if you don't have an employment contract that forbids it they can fire you or possibly even deduct it from your pay anyway (and this reason might not be one that allows you to collect unemployment insurance benefits - but you'd need to check with an expert on that). You can refuse to pay AT&T directly, and they can cancel the internet service - and your employer can then do the same as in the previous condition. Or you can choose to pay it - or ask your employer to split the cost over a few checks if it is rather high - and that's about it.
Like the cost of anything else you have to pay - from your own food to your computer, clothes, etc - it's best to just consider it your own "cost of doing business" and decide if it's still in your interest to keep working there, and for something to consider in future pay negotiations! You may also qualify for an itemized Employee Business Expense deduction from the IRS, but you'll need to read the requirements carefully and get/keep a receipt for such expenses.