Special & General Relativity Questions and Answers
If I passed the Sun at near the speed of light, it's mass would be higher, so why wouldn't the planets fall in?
Because you are not stating the problem correctly with respect to your reference frame. Not only would the Sun's mass appear larger, but all other factors would also scale in their appropriate ways to make what you see operate consistently. It's not much of an answer, but you need to do the relativistic math to see what is happening in detail.
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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the NASA Astronomy Cafe, part of the NASA Education and Public Outreach program.