How many pages can I continuously print without damaging the printer?

You should be fine!

In general the main thing that heats up in a laser printer is called the fuser. The fuser usually accounts for the majority of a laser printer's power usage. The heat of the fuser, under substandard conditions, can definitely cause damage to interior parts, however newer printers will usually have a good ventilation system to aid in heat dissipation. Another feature of newer printers is a power saving feature which will turn off the fuser and let it cool down. The printer only starts up again after proper working temperature is reached. You'll know if you have this feature or not If you've ever printed 100 pages or so. The stops are very noticeable but at least you know it's for safety reasons.

The HL 5340D specifically is from early 2009 I believe. It more than likely has these features and then some. You can even see the abundance of ventilation area they've given it on the sides:

Something else: I would be leery of printing that many pages in one batch unless you have just installed a high yield toner cartridge. What happens when the toner runs out in the middle of the job? Can you reprint only the bad pages?