When using Landsat 8 imagery to calculate NDVI, will the NIR and Red bands be affected by cloud cover?
In other words, will clouds produce a wrong NDVI calculation?
Yes, it affects the values of NDVI, and it may not give the desired results sometimes. Information from Wikipedia: Normalized Difference Vegetation Index provides some details about the effects of cloud and snow as follows:
clouds and snow tend to be rather bright in the red (as well as other visible wavelengths) and quite dark in the near-infrared
Which implies based on mathematical equation of NDVI applied on dense forest:
It can be seen from its mathematical definition that the NDVI of an area containing a dense vegetation canopy will tend to positive values (say 0.3 to 0.8) while clouds and snow fields will be characterized by negative values of this index.
You can read the above article to get more information about the effects of clouds on NDVI.
The formula for a vegetative index is
VI = CH2 - CH1 where
CH1 is in the visible band (0.58 - 0.68 um) and
CH2 is in the near infrared band (0.725 - 1.00 um).
The Normalized Difference Vegetation index (NDVI) is formulated as,
NDVI = (CH2 - CH1) / (Ch2 + CH1)
According to 
Vegetated areas will generally yield high values for either index [VI or NDVI] because of their relatively high near-IR relectance and low visible reflectance. In contrast, clouds, water, and snow have larger visible relfectance than near-IR reflectance. Thus, these features yield a negative index.
Satellite studies taken over multiple days can be utilized to eliminate the effects of cloud coverage on an area, as long as there is not cloud coverage for the entire observation period.
The clouds effect on NDVI is then eliminated by selecting
For each pixel ... the greatest value on any day during the 14-day period (the highest NDVI value is assumed to represent the maximum vegetation "greenness" during the period). This process eliminates clouds from the composite (except in areas that are cloudy for all 14 days).
 Remote Sensing and Image Interpretation. Lillesand. Kiefer. Chipman. 7th Edition.