Can a Boeing 747 take off fully loaded on 3 engines?
The short answer as detailed to some degree in previous answers and comments is that a 747 can takeoff if the engine fails after the V1 speed is reached but not if the failure occurs before reaching V1. Unfortunately, in my opinion, the owner of one of the answers has deleted his answer, which while perhaps imperfect did provide useful information, and perhaps as important will remove the comments.
It's best to understand the V1 speed within the context of a balanced field takeoff. See this Wikipedia article for a full explanation. The first two paragraphs are:

A balanced field takeoff is a condition where the accelerate-stop distance required (ASDR) is equal to the takeoff distance required (TODR) for the aircraft weight, engine thrust, aircraft configuration and runway condition.1 For a given aircraft weight, engine thrust, aircraft configuration, and runway condition, the shortest runway length that complies with safety regulations is the balanced field length.
The rejected takeoff initial actions speed V1, or critical engine-failure recognition speed (Vcef),is the fastest speed at which the pilot must take the first actions to reject the takeoff (RTO). At speeds below V1 the aircraft may be brought to a halt before the end of the runway. At V1 the pilot must continue the takeoff even if an emergency is recognized.

Let's translate this into a practical situation using a 747-200 with old steam gauges. Prior to engine start five plastic bugs would be positioned around the rim of the airspeed indicator: the first at V1, the second at VR, the third at V2, and the last two at speeds not relevant to our discussion. We're rushing down the runway at takeoff power and all goes well. When we reach V1, the PNF says &quot;V1&quot; and the PF takes his hand off the thrust levers (he had them there in case of an abort). When we reach VR, the PNF says &quot;rotate&quot; and the PF brings the yoke back briskly but smoothly to a pre-determined nose attitude (usually 12 or 13 degrees). When the landing gear is off the runway, you'll hear the gear down lock retract unlocking the gear up lever. When the PNF sees a positive climb rate on the vertical speed, he'll say &quot;positive rate&quot;, the PF will call &quot;gear up&quot;, the PNF will move the gear lever to up. When the airplane reaches V2 the PNF will say &quot;V2&quot;.
The important thing to remember here is that if there had been an engine failure after V1, the only difference would have been that the first person to know which engine had failed would have called &quot;engine failure #1&quot; say. On the old 3-crew airplanes, that would usually be the f.e. Even if the engine was burning, no action would be taken until after V2, and probably not until after you had a clean airplane.
Just as a matter of information, here are some V speeds for a 747-200 with P&amp;W JT9D-7Q engines for a Flaps 10 takeoff at 840,000 lbs:

V1 168 kts
VR 181 kts
V2 189 kts

Now for an abort. An abort below 80 kts is no big deal. Between there and 100 kts gets more interesting. Above 100 kts it is not to be taken lightly, and an abort at say, 150 kts, is a bonafide emergency. A friend of mine had to do one of those, and he blew all of the 16 tires that had brakes. In my view, if an engine failure occurs within a few knots either way of V1, you're better off flying.
To quote from an accident report in which I was involved in the investigation:

The bang and the loss of power occurred around V1 speed. Two seconds after the bang, all four engines were brought back to idle, and braking action was initiated. The thrust reversers were not deployed.
The aircraft came to a stop 300 m after the end of runway 20, above a railroad embankment. The aircraft was severely damaged; it broke in three parts.