[AllenG (OP)]

RS Ophiuchi is a binary system consisting of a red giant (RS Ophiuchi A) and a white dwarf (RS Ophiuchi B). 
RS Ophiuchi B steals matter from the red giant and periodically explodes as a type 1A super nova.

RS Ophiuchi B also spews jets of matter at near light speed.  According to Einstein as matter accelerates it gains mass.  Would the particles of the stream cause gravitational disturbances if they traveled through our solar system since they should have near infinite mass?

[lightarrow]

RS Ophiuchi is a binary system consisting of a red giant (RS Ophiuchi A) and a white dwarf (RS Ophiuchi B). 
RS Ophiuchi B steals matter from the red giant and periodically explodes as a type 1A super nova.

RS Ophiuchi B also spews jets of matter at near light speed.  According to Einstein as matter accelerates it gains mass.

No, that's incorrect. Mass doesn't change with velocity.
http://www.thenakedscientists.com/forum/index.php?topic=16789.0
http://www.thenakedscientists.com/forum/index.php?topic=21363.0

  Would the particles of the stream cause gravitational disturbances if they traveled through our solar system

Yes, because not only mass, but energy too (and stress, and momentum) compare in the Einstein's equation
http://en.wikipedia.org/wiki/Introduction_to_general_relativity

since they should have near infinite mass?

Maybe a great value, but certainly not "near infinite" []