Spearman correlation between two genes I have FPKM data for three genes like below:

     Samples      gene1 gene2 gene3
TCGA-2Y-A9GS-01A   0.9   7.4  1.0
TCGA-2Y-A9GT-01A   0.8   1.0  0.3
TCGA-2Y-A9GU-01A   0.6   2.0  0.2
TCGA-2Y-A9GV-01A   1.2   0.5  0.1
TCGA-2Y-A9GW-01A   3.8   2.1  0.4
TCGA-2Y-A9GX-01A   2.3   2.0  1.5


I used cor.test

cor.test(~ gene1 + gene2, data = df2, method="spearman", continuity=FALSE, conf.level=0.95)

    Spearman's rank correlation rho

data:  gene1 and gene2
S = 5686100, p-value = 6.083e-09
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.2984045 


I have a warning message which I didn't see before.

Warning message:
In cor.test.default(x = c(0.9, 0.8, 0.6, 1.2, 3.8, 2.3, 3.8, 0.4,  :
  Cannot compute exact p-value with ties


Do I need to care about this warning message? Is it good using FPKM data for correlation?

For plotting I used ggscatter.

ggscatter(data, x = "gene1", y = "gene2", 
          add = "reg.line", conf.int = TRUE, 
          cor.coef = TRUE, cor.method = "spearman",
          xlab = "gene1", ylab = "gene2")


Is this fine or I need to use any log for the scatter plot?
Since Spearman is a rank-based test, it relies on you being able to accurately decide on the ranking of your observations by some metric (usually the magnitude of the numbers). If two observations have identical values (are tied), then they cannot be definitively ranked. Since the ranks are not unique, exact p-values cannot be determined.

e.g. in your data, for gene 2:

                  gene2 rank
TCGA-2Y-A9GS-01A    7.4    1
TCGA-2Y-A9GT-01A    1.0    5
TCGA-2Y-A9GU-01A    2.0   3= 
TCGA-2Y-A9GV-01A    0.5    6
TCGA-2Y-A9GW-01A    2.1    2
TCGA-2Y-A9GX-01A    2.0   3=


By way of example, it's possible to do the test with either TCGA-2Y-A9GU-01A or TCGA-2Y-A9GX-01A ranked number 3 - meaning a definitive p-value cannot be determined.

If there are only a few ties, you probably don't need to worry too much. If your FPKMs (or better, TPMs) were calculated with greater precision, then ties would be less likely. Is there a reason you only have these numbers to 1 decimal place?

Log or linear scale doesn't make a huge difference (and neither is 'wrong'). If you choose to plot log counts, you need to consider what happens if you have zeroes in your data.