To develop an understanding of this subject that will remain intact after a few beers, it is necessary to start from the basic principles.
In order to take-off an aircraft must accelerate to VR at which point it must be rotated nose-up into the take-off attitude. It must then continue to accelerate to the lift-off speed (VLOF), at which point it will become airborne. After lift-off the aircraft must climb away from the ground whilst accelerating to at least the take-off safety speed (V2 ) by the time it reaches screen height. If no engine failures occur during the take-off run, the speed at screen height must at least V3, which is slightly higher than V2. All these processes must be achieved within the take-off distance available (TODR).
The distance required to reach each of the above speeds, depends upon the acceleration rate that can be achieved. The principal factors affecting acceleration rate (and hence distances required) are aircraft mass and thrust available. Increasing mass decreases acceleration rate, thereby increasing the distances required to take-off. Increasing thrust increases acceleration rate, thereby decreasing the distances required. But thrust is proportional to the mass flow rate of air passing through the engine. This in turn is proportional to air density. So anything which reduces air density reduces thrust and acceleration rate and hence increases the take-off distances required at any given aircraft mass. Factors which reduce air density and increase take-off distance required include increasing pressure altitude, increasing ambient temperature and increasing humidity.
In addition to the above factors, the failure of one or more engines during the take-off run will reduce thrust and acceleration rates, thereby increasing the take-off distance required. If the engine failure occurs early in the take-off run, the reduced acceleration rate will make it impossible to complete the take-off safely in the distance available. If however the failure occurs later, when the aircraft is closer to VLOF, the take-off can be completed safely. On the other hand an early decision to abort the take-off will enable the aircraft to stop safely within the remaining distance available. But a decision to abort made very late in the take-off run, is likely to result in the aircraft running into the weeds (or worse). So there is a point in the take-of run before which the take-off must be aborted if an engine fails, and another point beyond which it is safer to go on.
These two points represent (two of) the allowable limits of the take-off decision speed V1.
Now consider an aircraft taking-off from a very long runway at sea level on a cold dry day. Its thrust will be high and if it has a very small fuel load and no passengers or freight on board, its take-off mass will be low. This will enable it to accelerate quickly to a speed from which it can safely complete the take-off, even if an engine subsequently fails. This speed is called VGO, and is the minimum acceptable value for the take-off decision speed (V1). As the aircraft continues to accelerate down the runway it will eventually reach a speed at which it is just possible to abort the take-off and bring the aircraft to a full stop within the remaining distance available. This speed is called VSTOP and is the maximum allowable value for V1.
The values of VR, VLOF V2 and V3 are related to the dynamic pressure required to create lift. So if aircraft mass increases, this will increase the values of these speeds. This means that increasing take-off mass increases the speeds that must be attained at each stage of the take-off.
If a second take-off is then carried out at a slightly higher mass, the acceleration rate will be reduced, so a greater distance will be required to reach the increased VGO. This means that VGO will be achieved further down the runway. But the increased mass will also reduce the deceleration rate that can be achieved, so more distance will be required to stop the aircraft following a decision to abort the take-off. So the value of VSTOP will be lower, and will be reached earlier in the take-off run. This means that the values of VGO and VSTOP and the points on the runway at which each is reached will be closer together.
If further take-offs are carried each at a slightly greater take-off mass, a mass will be reached at which the values of VGO and VSTOP and the points on the runway at which they are reached are identical. There is then only one possible value for V1 which must be equal to both VGO and VSTOP. In this condition the aircraft is said to be at its Field Limited Take-off Mass (FLTOM).
If take-off mass is increased further, the value of VGO will be greater than that of VSTOP. This would mean that when taking off, the aircraft would lose the ability stop (at VSTOP) before attaining the ability to continue the take-off safely (at VGO). There would therefore be a period during the take-off run (between VSTOP and VGO) during which an engine failure would inevitably cause an accident. It is to avoid this situation that aircraft take-off mass must never exceed the field limited take-off mass.
If we now go back to the original question “will increasing mass increase V1?” The answer is yes, provided there is sufficient distance available to abort the take-off from this increased V1. In effect this means that the higher mass must not be greater than the field limited take-of mass.
There are of course various other factors that need to be taken into account but this post is already too long to mention them.