To find the number of moles of methane ($CH_4$) required to produce 11g of $CO_2$ upon complete combustion, we first need the balanced chemical equation for the combustion of methane:

$$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2O$$

From the balanced equation, we see that 1 mole of $CH_4$ produces 1 mole of $CO_2$. Therefore, the moles of $CH_4$ needed to produce a certain amount of $CO_2$ will be equal to the moles of $CO_2$ produced.

To find the moles of $CO_2$ produced from 11g of $CO_2$, we use the formula:

$$\text{Moles} = \frac{\text{Mass}}{\text{Molar mass}}$$

The molar mass of $CO_2$ is $44 \mathrm{~g/mol}$ (12 from carbon and 16$\times$2 from oxygen). Then, the moles of $CO_2$ produced from 11g of $CO_2$ can be calculated as:

$$\text{Moles of } CO_2 = \frac{11 \mathrm{~g}}{44 \mathrm{g/mol}} = 0.25 \mathrm{~mol}$$

Since the stoichiometry of the reaction between methane and oxygen is 1:1 for $CH_4$ and $CO_2$, the moles of $CH_4$ required to produce 11g of $CO_2$ is also 0.25 moles.

Therefore, the correct answer is Option B: 0.25.