“Bourse” is just another word for “exchange”, and the creation of one in Iran is an attempt to capture more of the economic activity associated with international oil markets and also perhaps to exert more control over oil markets.
The US gains directly from the fact that people hold US currency (since it costs almost nothing to print, but can be used by the US government to buy goods and services – this is called “seignorage”) and indirectly in terms of perceived power and influence from the fact that the $US is the dominant world currency. The switch to euros threatens both. However, the total benefits are not that great. The seignorage benefit to the US from overseas holdings of $US is between $15 billion and $50 billion per year, and the United States has many more important sources of power and influence than the $US.
There is a lot of talk, as in the Peak Oil article linked above, about how Saddam’s switch to euros led to the Iraq War. Actually, as the Iran example shows, overt decisions to switch are usually the result of bad relations with the US, not the cause. Still, this is part of a general pattern of incidents, small in themselves, where aggressive US foreign policy is exacerbating conflict over economic issues, and thereby weakening the US economy.