What are the ways we can avoid getting hit by Sniper's Ultimate ability, Assassinate? Which items and/or hero abilities can be used to escape it once it has locked on to a hero?
There're several ways to do this, most require low ping & fast reflexes:
In DotA, disjointing consist in removing oneself from being the target of a projectile. This can be achieved in several ways :
Blocking the damage
By Becoming invulnerable :
Invulnerability is granted by several skills (and Eul's Scepter) in the game. You can become invulnerable by
By Blocking the Spell completely
If you have a Linken's Sphere or an ally uses his to buff you, the spell will completely be cancelled.
Note that all these also apply to most single-target spells
Sniper's ultimate Assassinate is single target magic damage. Anything that blocks single target spells or makes you immune to magic damage will prevent it and it's damage can be reduced through magic resist such as cloak, hood, or pipe. The projectile, while fast, can be disjointed. Sniper can also be interrupted as it's a channeling ability prior to firing.
The request for specific hero abilities who can avoid it is somewhat unreasonable because there are so many. Look at the heroes you like to play and dig into their abilities. Anything that allows for disjoints (blinks, tornadoes, removals, invulnerabilities, slark's ult) can dodge the in air bullet with timing.
The quick list of items that can help you with the damage is: Hood, Pipe, Linkens, Euls, BKB, TP, Manta, Blink, Bloodstone, and finally if you're losing stack bracers.
You can :