This depends on your definitions of assets and liabilities.
The word "asset" has a fairly straight forward definition. Generally speaking, an asset in finance is something that you own/control that has economic value. The asset has value because it is generating income for you or because you expect that it will be worth something to someone in the future.
"Liability" is tougher to define, and depends on context. In accounting, a liability is a debt or obligation that is owed. It is essentially the opposite of an asset; where an asset represents something of value that you own, increasing your balance sheet, a liability is a value that you owe, decreasing your balance sheet.
In that sense, a website or domain name that you own is an asset, not a liability, because it is something you own that has some value. It is not a debt.
Many people use the word "liability" informally to refer to a bad asset: something that is losing value or is causing more in expenses than it is generating in income. (See definition #5 on Wiktionary.) With this definition, you might consider a website or a domain name a liability if it is losing money.
Alternatively, depending on your business, you might not consider it an asset or a liability, but an expense instead. An expense is a cost of doing business. For example, if your business is selling something, you might need a website to make that happen. The website isn't purchased as an investment, and it might not have any value apart from your business. It is simply a necessary expense for your business.