How can I simply load a greyscale tiff in libtiff and get an array of pixel intensities?

I think you should read Using The TIFF Library article. It contains enough information to get started with libtiff.

Here is some code to read image scanlines and print values of each sample.

main()
{
    TIFF* tif = TIFFOpen("myfile.tif", "r");
    if (tif) {
        uint32 imagelength;
        tsize_t scanline;
        tdata_t buf;
        uint32 row;
        uint32 col;

        TIFFGetField(tif, TIFFTAG_IMAGELENGTH, &imagelength);
        scanline = TIFFScanlineSize(tif);
        buf = _TIFFmalloc(scanline);
        for (row = 0; row < imagelength; row++)
        {
            TIFFReadScanline(tif, buf, row);
            for (col = 0; col < scanline; col++)
                printf("%d ", buf[col]);

            printf("\n");
        }
        _TIFFfree(buf);
        TIFFClose(tif);
    }
}

Regarding this article, I think it'll be better to use TIFFRGBAImage approach, because as it turned out TIFF file could be one of different formats: tiled, scanline-based and strip-oriented. Here's an example from the same article.

TIFF* tif = TIFFOpen(argv[1], "r");
if (tif) {
    uint32 w, h;
    size_t npixels;
    uint32* raster;

    TIFFGetField(tif, TIFFTAG_IMAGEWIDTH, &w);
    TIFFGetField(tif, TIFFTAG_IMAGELENGTH, &h);
    npixels = w * h;
    raster = (uint32*) _TIFFmalloc(npixels * sizeof (uint32));
    if (raster != NULL) {
        if (TIFFReadRGBAImage(tif, w, h, raster, 0)) {
            ...process raster data...
        }
        _TIFFfree(raster);
    }
    TIFFClose(tif);
}

raster is a uint32 array (maximum value= 0xffffffff) but you are trying to read a 16-bit array (maximum value 0xffff). you will run into 32bit to 16bit conversion problems. Reading the scanline method is the better way to do it. This way you can convert the void* buf to uint16* and access the pixel values.

#include <stdio.h>
#include <stdlib.h>
#include <iostream>
#include <inttypes.h>
#include "tiffio.h"


using namespace std;


void printArray(uint16 * array, uint16 width);
int main()
{


    TIFF* tif = TIFFOpen("16bit_grayscale_image.tif", "r");
     if (tif) {
    uint32 imagelength,height;
    tdata_t buf;
    uint32 row;
    uint32 config;

    TIFFGetField(tif, TIFFTAG_IMAGELENGTH, &imagelength);
     TIFFGetField(tif, TIFFTAG_IMAGELENGTH, &height);
    TIFFGetField(tif, TIFFTAG_PLANARCONFIG, &config);
    buf = _TIFFmalloc(TIFFScanlineSize(tif));


        uint16 s, nsamples;
        uint16* data;
        TIFFGetField(tif, TIFFTAG_SAMPLESPERPIXEL, &nsamples);
        for (s = 0; s < nsamples; s++)
        {
            for (row = 0; row < imagelength; row++)
                {
                TIFFReadScanline(tif, buf, row, s);
                data=(uint16*)buf;
                printArray(data,imagelength);
                }
                // printArray(data,imagelength,height);
        }


    _TIFFfree(buf);
    TIFFClose(tif);
    }
    exit(0);
}



void printArray(uint16 * array, uint16 width)
{
    uint32 i;
    for (i=0;i<width;i++)
    {
        printf("%u ", array[i]);
    }
        printf("\n");


}