Is my status as an approved asylee in danger if I marry an undocumented immigrant?

No law prohibits an approved asylee from marrying anyone he or she wishes to marry, including an undocumented immigrant.  The legal status of the asylee will remain in place and he or she can still apply for permanent residency after one year of continuous residence as an asylee in the United States. 

We're assuming, of course, that this is a valid marriage -- not just a fraud to help the undocumented immigrant obtain status in the United States. Committing marriage fraud is a crime, for which you could indeed be deported (removed) from the United States. But given how long it will take the asylee to help the immigrant obtain a green card, this would be a sadly ineffective way to attempt marriage fraud in the first place.

The asylee will not have much power to help the undocumented immigrant  get a U.S. green card. Because the marriage had not taken place when the grant of asylum occurred, the new spouse may not receive status as a derivative asylee.

What's more, the asylee will have to wait until attaining a green card to file a visa petition (Form I-130) for the immigrant, which will start the immigration process for that person -- but not give him or her any legal right to remain in the United States. And the wait to actually become visa-eligible could be a long one, because spouses of lawful permanent residents are in the 2A preference category, with annual limits on visas and a years-long waiting list. See "Does Filing an I-130 Form Grant Lawful Status in the United States?" for a full explanation.

Consult a lawyer for a personal analysis of whether it is possible to obtain legal status for the undocumented immigrant.