Will sticky tires kill a stock 10-bolt rear?

It depends which 10-bolt. Stickier tires put more shock into the axle and driveline, and Speedway's ratings drop as traction goes up:

GM 10-bolt	Stock capacity (Speedway)	Notes
8.5 / 8.6	The only high-performance 10-bolt; can hold about 1,000 hp built properly	Large square lugs at 5 and 7 o'clock; flat webs on the front; 28- or 30-spline
8.2 (1964–72)	About 400 hp on street tires, 350 on drag radials, 300 on slicks	Built for racing: about 500 hp on street tires, 400–450 on slicks
7.5 / 7.625	About 350 hp on street tires	Maxes out around 500 hp with maximum effort and expense; 28-spline only, no 30-spline upgrade

Telling them apart: measure from the lower center cover bolt to its neighbors — about 3-1/4 in on a 7.5, 3-3/4 in on an 8.5/8.6. The 8.5 has large square lugs at 5 and 7 o'clock; the 7.5's are small. Confirm with the codes stamped on the axle tube.