What are the consequences of inbreeding?
You are right. Inbreeding strongly increases overall homozygosity which subjects inbred individuals to diseases caused by rare recessive alleles. In non-inbred individuals the chance is quite low to receive those because many deleterious variants (and in fact, most segregating alleles we can observe) are recessive. Most often, but depending on the dominance coefficient, these 'hide' in healthy heterozygous carriers and when very closely related individuals breed (with a lot of variants that are identical by descent) there is a high chance that some of the 'hidden' deleterious variants are passed to the offspring.

You have to be aware that there are in fact two conceptually different processes that can be both be referred to as inbreeding but are also somewhat continuously linked (see below): 


Mating systems that allow offspring between actually related individuals (measured by the degree of relatedness), what in humans often is referred to as incest or consanguinity.
Populations with very low effective population size $N_e$ generally exhibit higher relatedness due to lack of genetic diversity (excess of homozygous sites). Therefore, populations with low $N_e$ are referred to as inbred even if no actually related individuals are mating.  


There is some evidence that inbreeding in the first sense is linked to cognitive abilities in humans:


Bashi (1977) investigated the effect of offspring having first-cousin (i.e. between children of siblings) and double first-cousin (i.e. between children of siblings and unrelated siblings, sharing as much variants as half-siblings but with more recombination events) parents while (at least somewhat) correcting for socioeconomic effects. He found an inbreeding depression with respect to cognition that could 


  result either from the general increase in homozygosity [...] or from decrease in performance resulting from homozygosity for specific recessive alleles (highlighted by me). The higher variance of the double cousin group in some of the tests favours the second interpretation.

Woodley (2012) presents evidence for slightly lower IQs caused by inbreeding, however, he also mentions that the effect is way smaller than socioeconomic effects:  


  Consanguinity could subtly influence IQ at larger scales as a result of small IQ handicaps bought about through inbreeding being amplified into much larger differences through their effect on factors that maximize IQ such as access to education and adequate nutrition.

Fareed and Afzal (2014) investigated verbal IQ, performance IQ, and full-scale IQ and found that all of these IQ parameters are significantly lower in inbred children compared to non-inbred children - actually the difference increases significantly with the degree of relatedness. They conclude that there is 


  evidence for inbreeding depression on cognitive abilities among children.



Please keep in mind that, for for sociological/ethical reasons, this is highly controversial, especially when the two concepts above are intermixed - human populations underwent differential and variably strong periods of low $N_e$, i.e. were subject to stronger or weaker inbreeding in the second sense. When reading the citation of Bashi (1977) above carefully, you will notice that he takes good care not to intermix those. Inbreeding in the first sense leaves large runs of homozygosity (ROH, blocks without heterozygous sites, i.e. a clustered and local lack of variation) whereas inbreeding in the second sense increases homozygosity in the genome less selectively. Therefore, the distribution of homozygous sites can be used to infer whether inbreeding is recent (first sense) or old (second sense) (see for example McQuillan et al. (2008) - here you see that both concepts form a continuum: where do you set the cutoff between recent and old? what is an appropriate threshold size for ROH? ...). Regardless, Bashi's findings indicate that effects of inbreeding are caused by recent inbreeding as he presents evidence that his results are rather driven by specific deleterious recessive variants than overall excess homozygosity. Finally, even though the study by Fareed and Afzal (2014) shows rather large effects of recent inbreeding in IQ measures, the results from Woodley (2012) show that one needs to be really careful to separate genetic from environmental components as his study suggests that the latter contribute more to the observed decrease in IQ.