Why spider's web don't burn when electricity touches it?
Spider silk can conduct electricity. Some spiders use this material property of conductivity to capture prey. https://www.ncbi.nlm.nih.gov/pubmed/24323174

So I'm going on the assumption that both electrical lines spanning the web are bare and that they are live, i.e. carrying current. I will also assume that these lines are carrying the same current.

With electricity, it might help to think of the analogy of a waterfall. Water at the top of a waterfall has potential energy, which is converted to kinetic energy when it goes over the waterfall. The water accelerates towards the earth as it falls. There is a height or "gradient" between the top of the waterfall and its bottom, which allows water to accelerate from the top towards the bottom of the fall.

Electrons in electricity work in much the same way. Between two points along a conductor, you need a voltage gradient to establish an electric field, which moves electrons in a particular direction — just as the waterfall creates a gradient of high ground to low ground that enables water to flow.

If you have two wires (both conductors) running at the same voltage, and if you connect them with another conductor, like a spider's web, electrons will not run through the web because there is no voltage differential that creates an electric field that drives the flow of electrons between that connection. https://en.wikipedia.org/wiki/Electric_potential

Here, the web and spider are likely unhurt, because the two lines do not provide a voltage difference needed to have electrons move across the silk, like water would flow over a waterfall.

If the spider web was grounded — if its web were instead connected from one of the electrical wires to the earth — electricity would likely run through the web and, potentially, burn the web (and the spider). But from your photograph, it looks like the web is connected to both wires that ℗r assumption) are at the same voltage.

This is the same principle that allows electricians to work on maintaining electrical pylons without shutting down the power grid. https://en.wikipedia.org/wiki/Live-line_working

They work in such a way as to maintain the same voltage as the lines they are working on. If they were grounded — if they held a copper wire, say, that was plugged into the earth — or if there was anything conductive that the electrician was connected to that established a dangerous voltage gradient, and if they started working on live electrical lines, they would get fried. Same with the spider, if the web was grounded in a similar way.

Further, if the two wires were running at different voltages, then electricity could conduct through the spider web and cause some damage.

This answer relies on a lot of assumptions from the photograph. Feel free to clarify if those assumptions would change the parameters of the question and my answer.