As neutrons are more massive than protons, does the Sun increase its mass while fusioning elements? Where is the mass coming from when neutrons are produced from protons in the Sun?
If a positron is made, will it possibly annihilate with an nearby electron?
 A: Stars fuse hydrogen (bare protons) to helium (two protons and two neutrons). But the reaction has a number of intermediate steps. The first step fuses two protons to deuterium, and a positron and a neutrino:
$$
    ^1_1H + ^1_1H ~ \rightarrow ~ ^2_1H + e^+ + \nu_e
$$
You are correct that a neutron has higher rest mass than a proton, but this reaction never generates a free neutron. The neutron is bound to the other proton from the moment it is generated. The binding energy compensates for the extra rest mass energy.
https://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain_reaction
A: No it is decreasing in mass. 
You must take into account binding energy in the atom core. The protons and neutrons are bound together really strongly by the strong nuclear force inside the core of a Helium atom. 
It takes lots and lots of energy to try to pull them apart. 
Firstly you must overcome electromagnetic force to push them together so closely they fusion. 
But at the scale of an atom core, the strong nuclear force is very much stronger, so once they fusion the electromagnetic force has no chance to repel the protons from each other.
A: While a free neutron does have more mass than a free proton, a bound helium-4 nucleus has less mass than two free protons and two free neutrons. In fact, the helium-4 nucleus has less mass than four free protons.  The difference goes into the binding energy of the nucleus. Therefore, as the other answers state correctly, stars are constantly losing mass, not gaining it, through their fusion reactions.
A: In fact, the Sun is losing mass all the time. It radiates large amounts of energy, and through the energy-mass relationship $E = m c^2$, radiating energy means radiating mass.
Since the mass of a helium atom is less than the mass of the four free protons which enter the fusion process, one can consider fusion the process of converting mass into energy.

A: Yes, the conversion from hydrogen to helium by nuclear fusion releases energy at the expense of the products having less mass than the reactants.  The linking equation between energy and loss of mass was proposed by Einstein, $E= \Delta m\,c^2$. 
The numbers are quite astronomical.   
Due to nuclear fusion the Sun loses about 4 million tonnes ($4\,000\,000\,000 \ \mathrm{kg}$) of mass every second and in the process produces $3.6 \times 10 ^{26} \ \mathrm J$ of energy per second.
This may seem a lot but if the Sun kept shining for another $5$ billion years it would have lost about $\frac {3}{10\,000}^{\text{th}}$ of its total mass.  
The other method of mass loss from the Sun is the solar wind, which are ionised particles "boiled off" from the surface of the Sun.
This process accounts for about $\frac 14$ the total loss of mass from the Sun.
A: The sun loses about 4 million tons per second in mass. This is largely in the form of light (which has a mass from E=mc^2), and charged particles such as helium nuclei.
A: Note that the question asked Where does the mass coming from when neutrons are produced from protons. A neutron is basically(*) a proton+electron. So the mass was already there as an electron and it basically merged with a proton to form the neutron.
As others correctly explained, when protons and neutrons bind into a nucleus they release energy. E=MC^2, the bound nucleus has lower mass than the free protons and neutrons. That energy comes out as sunlight.
(*)Footnote: For simplicity I deliberately ignored neutrinos.
