We present correlations between 9 CO transition ($J=4-3$ to $12-11$) and beam-matched far-infrared (Far-IR) luminosities ($L_{mathrm{FIR},,b}$) among 167 local galaxies, using {it{Herschel}} Spectral and Photometric Imaging Receiver Fourier Transform
Spectrometer (SPIRE; FTS) spectroscopic data and Photoconductor Array Camera and Spectrometer (PACS) photometry data. We adopt entire-galaxy FIR luminosities ($L_{mathrm{FIR},,e}$) from the {it{IRAS}} Revised Bright Galaxy Sample and correct to $L_{mathrm{FIR},,b}$ using PACS images to match the varying FTS beam sizes. All 9 correlations between $L_{mathrm{CO}}$ and $L_{mathrm{FIR},,b}$ are essentially linear and tight ($sigma$=0.2-0.3 dex dispersion), even for the highest transition, $J=12-11$. This supports the notion that the star formation rate (SFR) is linearly correlated with the dense molecular gas ($n_{mathrm{H}_2}gtrsim10^{4-6},cm^{-3}$). We divide the entire sample into three subsamples and find that smaller sample sizes can induce large differences in the correlation slopes. We also derive an average CO spectral line energy distribution (SLED) for the entire sample and discuss the implied average molecular gas properties for these local galaxies. We further extend our sample to high-{it{z}} galaxies with CO $J=5-4$ data from the literature as an example, including submillimeter galaxies (SMGs) and normal star-forming BzKs. BzKs have similar FIR/CO(5-4) ratios as that of local galaxies, an follow well the locally-determined correlation, whereas SMG ratios fall around or slightly above the local correlation with large uncertainties. Finally, by including Galactic CO($J=10-9$) data as well as very limited high-{it{z}} CO $J=10-9$ data, we verify that the CO(10-9) -- FIR correlation successfully extends to Galactic young stellar objects, suggesting that linear correlations are valid over 15 orders of magnitude.
