Do you lose any gold by listing a purchase offer, then removing it?

No.

In the Trading Post, all fees are always applied to the seller.

When creating a sell order, the seller pays:

• a 5% listing fee upon listing, and
• a 10% exchange fee upon sale.

When filling a buy order, the seller pays:

• a 5% listing fee (technically listed for zero seconds), and
• a 10% exchange fee upon filling the order.

In all situations, the buyer is entirely free of Trading Post fees, including placing a buy order that is never filled. This can be seen by placing a buy order, immediately cancelling it, and then checking your gold total after getting your money back from a Black Lion Trading Company NPC. So, if you have outstanding buy orders and are in need of your money back, you can simply get it back without any loss.

It's worth noting that this doesn't mean you should remove your buy orders frequently. Doing so could cause you to miss out on a deal.

Once you place your buy order, you are placed in a queue for your selected price. If you place a buy order for 1s 60c and there are currently 5 buy orders for 1s 60c, then yours will be the 6th buy order to be filled. For items with a lot of buy orders, removing your buy order will remove you from this queue if you ever decide to attempt to buy at the same price again. For high-priced items, you could just increase your buy order slightly and get a similar spot in the queue. But for lower-priced items, there's not as much space to squeeze your way in.

Yes you lose gold by removing items and placing them again. I just tested and lost gold.