Does risk aversion of utility function cause the existence of positive interest rate? In standard macro model, it is usually time preference that causes positive interest rate. But is there anything to do with risk aversion of utility function that causes existence of positive interest rate?
Yes and no; it depends on which interest rate you look at.

You are right in that risk aversion affects interest rates, but the direction can go both ways. In what follows I look at an economy with risky (stochastic) and non risky assets and risk averse agents. For the thought experiment, we increase the volatility of the risky asset, "increasing its riskiness".

Risky Asset

Despite having the same expected return, due to Jensen's inequality, the risky asset gives less utility to the agents. The risky asset will need to pay a higher interest, a risk premium, to attract investors.

Safe Asset

We have increased the aggregate risk of the economy by making the stochastic asset more volatile. Agent's added utility from insurance (through safe assets) has increased. Demand for safe assets at the old interest increases. Holding the supply of safe assets fixed, this means that the safe asset's interest rate will go down.

To understand this fully, you might want to write down a two period model with two different assets, one standard normal with standard deviation $\sigma$ and one deterministic. Examine how interest rates for both assets change if you increase $\sigma$.