Old star vs New star We know that with the passage of time, hydrogen inside (near or at the core) a star is converted to helium and other heavy elements via fusion. We can then deduce that the old sun contains heavier elements in comparison to the new one. Does that mean that the mass of the old sun is more than the new sun?
Let me try to add some numbers to Steve's answer.

The Sun's luminosity is about $L_{\odot}=4\times10^{26}\text{ J/s}$. Now, if we assume that the majority of that energy comes from nuclear fusion, we have
$$L_{\odot}=\frac{dE}{dt}=\frac{d(mc^2)}{dt}=c^2\frac{dm}{dt}=c^2\dot{M}_{\text{nuc}}$$
Therefore, we can write the mass-loss due to nuclear fusion as $\dot{M}_{\text{nuc}}=L_{\odot}/c^2\sim4\times10^{9}\text{ kg/s}$.

Now, the Sun also loses mass via the solar wind. I believe the mass loss rate is a few times about $\dot{M}_{\text{wind}}\sim3\times10^{-14}M_{\odot}/\text{yr}\sim2\times10^9\text{ kg/s}$. Coronal mass ejections, solar flares and similar phenomena are often accounted for in this figure; at any rate, they don't add a significant amount.

This figure should increase as the Sun moves into the red giant (and especially asymptotic giant branch) phase of its life; it winds will increase to become much more significant that they are today. However, fusion will also increase, and the Sun's luminosity will increase by a factor of 1000 or 10000; therefore, the mass loss due to fusion will also increase. However, I think mass loss due to winds will increase more significantly, possibly by a factor of $\sim10^6$ - but I'll check that.

At any rate, an older star will have a lower mass than a younger star. For a Sun-like star, this would be hard to measure over the span of a human lifetime - remember that it only loses about $10^{-13}$ solar masses each year. Long-term records could make it possible, though, through various observations of the Sun and its effects. Centuries or millennia of data-gathering do tend to yield results.