A sample of 34 dwarf irregular galaxies (dIs) in the Local Volume, most nearer than 5 Mpc, has been imaged in the near-infrared (NIR) in J and Ks at the Canada-France-Hawaii Telescope (CFHT) in Hawaii and the Observatorio Astronómico Nacional in the Sierra San Pedro Mártir, in Mexico. Absolute magnitudes in Ks range from -14 to -18. In the CFHT images, stars brighter than MKs~-7.5 were resolved. We show that the resolved component comprises more than 50% of the light from star formation bursts within the last 3 Gyr. In most cases, the resolved population down to MKs=-7.5 represents less than 5% of the total NIR flux in Ks, with fractions in J being 1.5-2 times larger. Thus, the NIR light of dIs can be considered to be predominantly contributed by stars older than about 4 Gyr. Although exponential at large radii, surface brightness profiles for the unresolved component flatten in the centers. They can be fitted across the whole range of radii with a hyperbolic secant (sech) defined as a function of two parameters: the central surface brightness and the scale length of the exponential. With respect to this model, only two galaxies (NGC 1569 and NGC 3738) show an excess of flux in the center, both of which are hosting starbursts. Isophotal, total, and fitted sech magnitudes have been calculated for all galaxies for which the unresolved component was detected, along with semimajor axes at μJ=23 mag arcsec-2 and μKs=22 mag arcsec-2. The scale length and the semimajor axes correlate linearly with absolute isophotal magnitude. The same is true for colors and the central brightness. More luminous dIs tend to be larger, redder, and brighter in the center. The fraction of light contributed by young stars is independent of both luminosity and central surface brightness. The Tully-Fisher relation shows considerable scatter, but residuals are tied to surface brightness. The galaxies appear to lie in a ``fundamental plane'' defined by the sech absolute magnitude, the sech central surface brightness, and the H I line width. The rms of residuals in MK is only 0.4 mag, which implies that the plane can be used to evaluate the distances of star-forming dwarfs. Corrections for tilt do not reduce the residuals, so line widths must be governed predominantly by random motions. Color-magnitude diagrams (CMDs) are presented for 29 galaxies in which stars were resolved. Most show a finger centered around J-Ks=1 mag. In some cases, there is a red tail extending to J-Ks=2.5 mag. Most color profiles constructed for the unresolved component show a remarkably constant J-Ks=0.8-1.0 mag, matching the color of the finger in the CMDs. Based on observations obtained at the Canada-France-Hawaii Telescope, which is operated by the National Research Council of Canada, the Centre National de la Recherche Scientifique de France, and the University of Hawaii; also based on data acquired at OAN-SPM in Mexico.