So I am not a expert with team city but know java & tomcat and the issues of different java major versions.
So lets first find out what is where
[root@kick73 java]# cd /usr/java
[root@kick73 java]# ls -l
lrwxrwxrwx 1 root root 16 Apr 12 23:35 default -> /usr/java/latest
drwxr-xr-x 8 root root 233 Apr 12 23:35 jdk1.7.0_101
lrwxrwxrwx 1 root root 22 Apr 12 23:35 latest -> /usr/java/jdk1.7.0_101
Here you see that latest & default point to jdk1.7.0_101
Next check to see if /usr/bin/java points to /usr/java/default
[root@kick73 java]# ls -l /usr/bin/java
lrwxrwxrwx 1 root root 26 Apr 12 23:35 /usr/bin/java -> /usr/java/default/bin/java
So fare we are ok.... Now lets download latest jdk for Java 1.8.0.121. so go to https://java.com/en/download/manual.jsp#lin and select one of the middle 2 linux files depending on your machine arch. This should pull down a *.tgz file. If you get the RPM file that will not work well for you.
So transfer that to your host and lets load it into /opt (so as root)
mkdir /opt
cd /opt
cd /opt
extract the jdk using tar command
tar xvfpz jre-8u121-linux-x64.tar
So now you have java 7 in /usr/java and java 8 in /opt
Now go to where your tomcat is installed and go to bin directory. In my example my tomcat is /apps/apache-tomcat-8.0.43
cd /apps/apache-tomcat-8.0.43/bin
Next edit the file catalina.sh and the beginning has the following
JRE_HOME=/opt/jre1.8.0_121
CATALINA_HOME=/apps/apache-tomcat-8.0.43
JAVA_HOME=/opt/jre1.8.0_121
!/bin/sh
Change JRE_HOME and JAVA_HOME to point to the above example