Can confidence interval of positive values be negative if I use sample mean
+- sample standard deviation?
If I need a log-log plot (with confidence interval), what should I do in this case?
The first question has a simple answer: yes.
I interpret your question to mean, "can a strictly positive sample (where all data points are positive) have a 68% confidence interval for the normal distribution with a negative lower bound?"
Proof by construction: Let $X = [1, 1, 7]^T$. The mean is $3$, the sample standard deviation is $3.46$, so the 68% confidence interval is $(-0.464, 6.46)$. $\square$
One consequence of a confidence interval that includes zero is that we are unable to reject the hypothesis that the true population has a normal distribution with mean zero.
Confidence intervals are defined relative to a particular distribution. Defining a confidence interval as mean +/- sample standard deviation implicitly assumes a normal, or at least symmetric, distribution. If we choose a distribution that itself is non-negative, say $\chi^2$ or Poisson, then the confidence interval will never go below zero. It will, however, be asymmetric.
Plotting the confidence interval on a log-log plot is less clear cut. I can see how the problem arises: a log-log plot is completely appropriate for strictly positive data, but there's not one obvious way to plot the confidence interval. Here are some options, it roughly increasing order of difficulty: