When is it appropriate to select models by minimising the AIC? It is well established, at least among statisticians of some higher calibre, that models with the values of the AIC statistic within a certain threshold of the minimum value should be considered as appropriate as the model minimizing the AIC statistic. For example, in [1, p.221] we find 

Then models with small GCV or AIC would be considered best. Of course one should not just blindly minimize GCV or AIC. Rather, all models with reasonably small GCV or AIC values should be considered as potentially appropriate and evaluated according to their simplicity and scientific relevance.

Similarly, in [2, p.144] we have

It has been suggested (Duong, 1984) that models with AIC values within c of the minimum value should be considered competitive (with c=2 as a typical value). Selection from among the competitive models can then be based on such factors as whiteness of the residuals (Section 5.3) and model simplicity.

References: 


*

*Ruppert, D.; Wand, M. P. & Carrol, R. J. Semiparametric    Regression, Cambridge University Press, 2003   

*Brockwell, P. J. &    Davis, R. A. Introduction to time-series and forecasting, John Wiley    & Sons, 1996


So given the above,  which of the two models below should be preferred?
print( lh300 <- arima(lh, order=c(3,0,0)) )
# ... sigma^2 estimated as 0.1787:  log likelihood = -27.09,  aic = 64.18
print( lh100 <- arima(lh, order=c(1,0,0)) )
# ... sigma^2 estimated as 0.1975:  log likelihood = -29.38,  aic = 64.76

More generally, when is it appropriate to select models by blindly minimizing the AIC or related statistic?
 A: You can think of AIC as a providing a more reasonable (i.e., larger) $P$-value cutoff.  But model selection based on $P$-values or any other one-variable-at-a-time metric is frought with difficulties, having all the problems of stepwise variable selection.  Generally speaking, AIC works best if used to select a unique single parameter (e.g., shrinkage coefficient) or to compare 2 or 3 candidate models.  Otherwise, fitting the entire set of variables in some way, using shrinkage or data reduction, will often result in superior predictive discrimination.  Parsimony is at odds with predictive discrimination.
A: Paraphrasing from Cosma Shalizi’s lecture notes on the truth about Linear
Regression, thou shall never choose a model just because it happened to minimise a statistic like AIC, for

Every time someone solely uses an AIC statistic for model selection, an angel loses its
wings. Every time someone thoughtlessly minimises it, an angel not only loses its wings,
but is cast out of Heaven and falls in most extreme agony into the everlasting fire.

A: I would say it is often appropriate to use AIC in model selection, but rarely right to use it as the sole basis for model selection. We must also use substantive knowledge.
In your particular case, you are comparing a model with a 3rd order AR vs. one with a 1st order AR. In addition to AIC (or something similar) I would look at the the autocorrelation and partial autocorrelation plots. I would also consider what a 3rd order model would mean. Does it make sense? Does it add to substantive knowledge? (Or, if you are solely interested in prediction, does it help predict?)
More generally, it is sometimes the case that finding a very small effect size is interesting. 
