Sun constantly converts mass into energy, will this cause its gravity to decrease? If the sun is constantly converting the mass into energy, then will its gravitational field continue decreasing?
If the sun is constantly converting the mass into energy, then will its gravitational field go on decreasing?


It's a very interesting question and the answer is yes!

The solar constant indicates the mean solar radiation of electromagnetic waves (mostly in visible and near infrared light and I'll answer based on that. 

While the conversion of mass matter† to energy in the Sun's core now represents a loss of mass proper matter, it turns out that that energy (trapped in the Sun and slowly diffusing towards the surface) will have the same gravitational attraction as the matter it came from until it actually escapes the Sun!

There is some prompt mass and energy loss via neutrinos and it's significant, perhaps several hundred keV per neutrino I simply don't know the number yet. I'll ask a separate question about it. I'm guessing that losses due to the stellar wind are small, but I'll update here as soon as the following is answered:


How much mass does the Sun lose as light, neutrinos, and solar wind?


update: the answer there is that loss via neutrinos is only about 2.3% of the radiative loss, and on average loss via solar wind and coronal mass ejections is about 4E+16 kg/year, or about another 30% relative to the radiative loss described below.

The value $I$ is about 1360 Watts per square meter at $R$ = 1 AU which is about 150 million kilometers or 150 billion meters. So the total energy lost per second $P$ is 

$$P = 4 \pi R^2 I$$

Taking the time derivative of $E = m c^2$ we get

$$\frac{dE}{dt} = P = \frac{dm}{dt} c^2$$

so 

$$  \frac{dm}{dt} = \frac{1}{c^2} \ 4 \pi R^2 I$$

That means that the value of the mass that we use to calculate the Sun's gravitational attraction changes by about 4.3E+09 kilograms per second, or 1.3E+17 kilograms per year.

The Sun's current mass is about 2.00E+30 kilograms, so this effect changes by a very tiny fraction per year, about 6.7E-14.  Over the age of the Earth of 4.5 billion years, that's 3E-04, or about 0.03% if the Sun's output were constant. It has probably changed over this time of course, so this is just a rough estimate.

†Thanks to @S.Melted's answer for clarifying this.



added to my answer by @Tosic:


  The Earth feels no torque from any force during this (the force from radiation is radial), which means its angular momentum is conserved. This means
  $$R_1 v_1 = R v$$
  $$R_1 \sqrt{\frac{GM_1}{R_1}} = R \sqrt{\frac{GM}{R}}$$
  $$M_1 R_1 = M R$$
  We can see the Earth's orbital radius would change by a factor of 0.03% as well (M1 and M are solar masses).