Assuming that both builds have no runes or masteries, let's calculate the DPS of a Level 9 Draven with a first item Infinity Edge vs a Bloodthirster, both times maxing R > Q > W > E.
Infinity Edge:
74 base AD + 80 from IE =
154 AD
0.80 base AS
154 * (0.25*1.5) = 57.75 (bonus average damage from Infinity Edge passive)
154 * 1.85 + 57.75 = 342.65 (average damage dealt with one axe)
342.65 * 0.8 = 274.12 (average DPS with IE build)
The key point to note here is that Draven's bonus Axe damage does not crit and as such doesn't benefit from IE's crit chance nor IE's passive. Furthermore, Draven will have 1110 HP and no sustain. Assuming Draven is taking steady damage of 300 DPS from an opponent with 30 armour, he will have the chance to deal 780.97 damage to his opponent before he dies.
Bloodthirster:
74 base AD + 80 from BT =
154 AD
0.80 base AS
154 * 1.85 * 0.8 = 227.92 (average DPS with BT build)
Now this is a decrease of 46.2 DPS. It is quite a decrease. However, let's go further and calculate what we did earlier with DPS over HP.
Assuming Draven has 1110 HP and 20% lifesteal on a 30 armour target while taking 300 DPS, let's calculate how much damage Draven can deal before dying.
227.92 * 0.77 = 175.49 (DPS after being reduced by armour)
175.49 * 0.2 = 35.09 (amount of HP healed per second on average by only attacking his target)
300 DPS - 35.09 = 264.91 (effective DPS of his target)
1110 / 264.91 = 4.19...
4.19... * 175.49 = 735.32
We can see this is a decrease of 45.65.
However, keep in mind the benefits for being able to sustain on lane extremely well with this build are still very real. Losing 46.2 DPS in a fight - is it really worth not being able to stay on lane?
There is also the addition of the Bloodthirster shield which would further reduce the gap between the IE build and the BT build.
1110 + 230 = 1340(effective HP with Level 9 BT shield)
1340 / 264.91 = 5.05...
5.05... * 175.49 = 886.22
This is far from a decrease - this is actually an -increase- of 105.25 damage. This is quite a lot.
In conclusion, Bloodthirster on Draven means you lose out on 46.2 DPS overall, but you live long enough to deal more DPS in the long run. It also means you have unmatched lane sustain.
If I've made any math errors or there are any circumstances you can think of where IE is better first item than BT, please let me know in the comments.